A Guerra dos Cinco Reis teria ocorrido se Ned tivesse recusado o pedido de Robert?

Would the war still have occurred?

I think that it is highly likely that the end result would be similar, it might just happen with different players.

Remember who started this whole wheel in motion... Petyr "Chaos is a Ladder" Baelish! Considering how long he has been planning this plot it is likely that he is willing to change tactics and find the best result amid the chaos.

Also remember that Cersei is an active player in this game as well. She has been looking for a way to knock off good ol' Bobby B. The true start of the War is the suggested illegitimacy of Robert's children. Littlefinger would surely keep pushing this agenda to both Stannis and Renly once Robert is dead. Stannis also knew of the bastards as he was helping Jon Arryn investigate. While he fled to Dragonstone after Arryn's death, he was most likely trying to rally some support before pressing his claim.

No doubt Varys would want something similar to this out come as well. To get a Targaryen back on the throne the Seven Kingdoms need to be in a weakened state since the "invading" Targaryen army would have a much smaller, foreign army and no dragons this time around. The Lannisters are no easy foe as they control much of the kingdom's wealth.

Overall it was not a matter of IF it was a matter of WHEN. All the masterminds (Petyr Baelish, Varys, and Cersei) seem to be willing to bide their time and adjust to external events accordingly.

Would Robb have been crowned King in the North?

This is not as likely. If the Starks were brought into the new scenario (as I outlined above) Ned would have raised his banners for Stannis.

Ned refused Renly's offer while still in King's Landing because Stannis "has the right". But this is why Robb was proclaimed as King in the North in the first place, as basically since there was no clear heir to throne, why should they have one at all. The North was seeking justice/vengeance for the Lannisters beheading their liege lord. With Ned still alive, they are fighting to restore proper law with Stannis. Ned would not have allowed his bannermen to declare him King.

If Ned was killed in battle there would be no cause for vengeance from the North. Robb would carry on the good fight in the name of Stannis and either that side is victorious or is defeated.

Sim.

This is just like the famous fan argument "Would the war have broken out if Catelyn hadn't captured Tyrion?".

War of the Five Kings was going to happen regardless to the trigger. It would have been for the following reasons:

1. Lysa still would have written to her sister about Lannister role in murder of Jon Arryn and Catelyn would have told Ned who would then be further disposed to take the field against the Lannisters if it was true.
2. Robert still would have been murdered by Cersei as he was growing restless with time and Cersei was growing nervous. Plus to add to her anxiety, Renly was working to have Robert set Cersei aside and take a new Tyrell Queen. We can't have that now, can we?
3. Stannis would still have pressed his claim as he already knew that the Princes and the Princess were bastards of Jaime Lannister.
4. Renly still would have pressed his "claim" as he felt no qualms pressing it when he had no idea about the incestuous scandal. He wanted to be King regardless. Not to mention he had Loras Tyrell and therefore the Reach. With two Kingdoms in his pocket, he was going to start as the strongest candidate which he did.
5. Lannisters and Baratheons still would have clashed due to reason 2,3,4.
6. Given their proximity to King's Landing and Westerlands (And alliance with two great Houses the Arryns and Starks), Tullys would have been drawn into the fray by either the Baratheons or the Lannisters, not necessarily as an enemy but rather an ally of one party and therefore enemy of the other (Or knowing Tywin, as an enemy, hit hard first and take the initiative, blocking the entry of Tully allies into operations area as they would be predisposed to join Baratheons instead of Lannisters).
7. Any attack on Tullys would have dragged the Starks (And Arryns too theoretically) into the war.
8. If everything else failed, Littlefinger, the man pulling the strings, would have found another way to drag the Starks into the war.
9. Varys would have still kept peddling, and found a way to bring a war to weaken the Great Houses to push "Real Heirs" onto the throne.
10. Balon Greyjoy would have still launched his invasion once the war started. The old man sensed that a war was coming and he was determined to make the most of it. He had given up on Theon, as evident by the fact that when Theon came, Ironborn banners were already gathering. And Balon already considered Asha his "son" and heir. He had a faint hope that Theon might still be suitable but he was disappointed when he took a look at his last living son.

The war was inevitable. Tensions already existed and so did the suspicions. The only thing open to debate was, how would it start? And it began with Catelyn arresting Tyrion.

For detailed analysis of the stated reasons, view the linked answers.

As for Robb claiming the Winter throne, that could not have been foreseen. For most of the early fighting, The Northmen and Rivermen fought to secure release of Eddard Stark and go home. When Joffrey executed Eddard Stark, it became clear for Robb that peace with Joffrey was now impossible. He however still acknowledged that Tommen Baratheon had the right to throne and after Joffrey, He would ascend to the throne. It was Lord Umber who gave the stunning counsel that they should declare independence and let Tommen keep his throne, instead of bowing to another Southerner. Robb liked this idea and was crowned King. See the linked answer for detailed event.

Honestly, all we have to go by as far as an official answer is from George R. R. Martin himself, where he says:

NED AND THE NIGHT’S WATCH
If Ned had taken the black...

... it would have been a different novel
^{-So Spake Martin, 2012-04-22: Ned and the Night's Watch.}

That's it, that's all the he says on this topic. This is because getting into the specifics of "what would have happened if..." is just total pointless since as GRRM puts it "it would have been a different novel". Essa is the story that the author chose to write, and that's all that matters. Anything else is fan-fiction.

He also explains on another related topic (which is about Sansa's role in Ned's downfall, but illustrates this point brilliantly):

REGARDING SANSA Your question re Sansa...

The way I see it, it is not a case of all or nothing. No single person is to blame for Ned's downfall. Sansa played a role, certainly, but it would be unfair to put all the blame on her. But it would also be unfair to exonerate her. She was not privy to all of Ned's plans regarding Stannis, the gold cloaks, etc... but she knew more than just that her father planned to spirit her and Arya away from King's Landing. She knew when they were to leave, on what ship, how many men would be in their escort, who would have the command, where Arya was that morning, etc... all of which was useful to Cersei in planning and timing her move.

Ned's talk with Littlefinger was certainly a turning point, though I am not sure I would call it =the= turning point. There were other crucial decisions that could easily have changed all had they gone differently. You mention Ned's refusal of Renly, which was equally critical. And there is Varys to consider, as well as the minor but crucial player everyone forgets -- Janos Slynt, who might have chosen just to do his duty instead of selling the gold cloaks to the highest bidder.

So... all in all, I suppose my answer would be that there is no single villain in the piece who caused it all, but rather a good half dozen players whose actions were all in part responsible for what happened.

Hope that helps.

(And let me add that I am always astonished to be reminded how fiercely some of my readers argue these points. It's gratifying to know I have readers who care so much, although if truth be told sometimes I get the scary feeling that you people know these books better than I do... )
^{-So Spake Martin, 1999-10-04: Regarding Sansa.}

I've highlighted the key parts. Yes, he's talking about Sansa's involvement, but he is also talking about how his story is so in-depth that you can't simply pin actions and circumstances on one single event or person. All things are intertwined.

A história se repete

Look, the history of Westeros (and the whole planet for that matter) has been cyclical. In that they are constantly repeating the past. What with their wars, their prophecies, their beliefs and their family pride.

Let's take a look at the history of the Brackens e a Blackwoods. Who, as Jaime Lannister finds out whilst talking to Hoster "Hos" Blackwood in Uma dança com dragões, have been warring for many centuries, and probably don't even know or remember the real razões:

..."How did all this [enmity] begin, between Blackwood and Bracken? Is it written down?"
"It is, my lord," the boy said, "but some of the histories were penned by their maesters and some by ours, centuries after the events that they purport to chronicle. It goes back to the Age of Heroes.

...

"So you are fighting over a crown that one of you took from the other back when the Casterlys still held Casterly Rock, is that the root of it? The crown of a kingdom that has not existed for thousands of years?" He chuckled. "So many years, so many wars, so many kings … you'd think someone would have made a peace."
"Someone did, my lord. Many someones. We've had a hundred peaces with the Brackens, many sealed with marriages. There's Blackwood blood in every Bracken, and Bracken blood in every Blackwood. The Old King's Peace lasted half a century. But then some fresh quarrel broke out, and the old wounds opened and began to bleed again. That's how it always happens, my father says. So long as men remember the wrongs done to their forebears, no peace will ever last. So we go on century after century, with us hating the Brackens and them hating us. My father says there will never be an end to it."

^{-A Song of Ice and Fire: Book Five - A Dance With Dragons, Chapter Fourty-Eight (Jaime I).}

And so it is with the War of the Five Kings, as you can see that there are many 'belligerents' each with their own, likely as justified, reasons, most of which have nothing to do with Ned (as adapted from my answer on Quanto tempo durou a Guerra dos Cinco Reis?):

The O.G. Five

• Robb Stark:
  • Wanted freedom for The North, Riverlands and The Iron Islands
• Stannis Baratheon:
  • Felt the Throne was his by rights
• Joffrey Baratheon:
  • Felt the Throne was his by rights
• Balon Greyjoy:
  • Wanted freedom for the Iron Islands, so that they could rape, pillage and be total douche-bags in peace.
  • Also wanted to annex The North, because, why not?
• Renly Baratheon:
  • For the lulz

You can't just say that just because the catalyst was Ned's beheading desta vez that the War wouldn't have gone ahead if that didn't happen.

Não necessariamente

On his death bed, Robert Baratheon wanted to explicitly name Joffrey as his heir. Ned Stark substituted language about the rightful heir. Someone else might have written the passage as Robert intended. If Robert had explicitly said that Joffrey was his heir, then that would have greatly reduced the Stannis claim.

It has been argued that Robert's will did not matter because only Cersei read it. Even if true, Ned announced it in open court, the one place guaranteed to have active spies for all factions. Beyond that, if Cersei had had a will legitimizing Joffrey, she would have treated it differently. She'd have published it rather than destroying it. That might not have mattered to Stannis, but it would have mattered to the Tyrells and various Baratheon allies.

Without the tension between Stannis and Joffrey, there wouldn't have been room for Renly to press his claim. In particular, without Ned's claims of Joffrey's illegitimacy, Renly wouldn't have been able to run as the alternative to his brother. The Tyrells still might have preferred Renly to Stannis, but without Ned's public announcement of the illegitimacy, they would have little reason to prefer Renly to Joffrey. Much easier to marry Joffrey to Margaery.

Robert wanted Joffrey and Sansa to marry (possibly a Littlefinger suggestion). But Robert would have been dead. The Lannisters would have dumped that marriage if it gave them an advantage. We know this because that is what actually happened after Renly died. They put aside the Joffrey/Sansa engagement in favor of marrying Joffrey and Margaery. They still could have offered a Tommen/Arya betrothal or a Robb/Myrcella marriage to the Starks. The Tyrells could have sweetened that with a Loras/Sansa marriage.

Renly might have wanted to be king, but he wouldn't have been able to offer himself as a claimant without the illegitimacy. He needed the honest stamp that Ned gave it to make people seriously consider the possibility of a King Stannis. He had an in with the Tyrells through Loras, but they would likely have taken the easier path. Without Ned as Hand, it would have been far easier to negotiate a Margaery/Joffrey marriage.

Robb Stark wouldn't have been head of House Stark much less King of the North. Ned might have supported Joffrey based on Robert's expressed wishes. Note that Stannis did not try to recruit Ned. He left that to agents that stayed in King's Landing.

Ned had two ways to control the Greyjoys. First, he could threaten the hostage, Theon. Second, he had an army next door. It's noteworthy that the first place that the Greyjoys attacked was the North. Robb only had Theon, as he took the army south. If Greyjoy had rebelled, both Joffrey and Stannis would have agreed that Ned should put down that rebellion.

Balon Greyjoy simply wouldn't have rebelled under that circumstance. He needed Robb to take the entire Northern army south to make an opening. Ned wouldn't have taken the whole army south to support either Joffrey or Stannis. He'd have left behind enough soldiers to avoid an invasion from the Iron Islands. Or he'd have simply stayed put. Further, Balon rebelled after Robb did.

Robb rebelling meant that people would treat with Balon, as losing the Iron Islands would have hurt less than losing the North. No rebellious Robb, no room for Balon to split off as the small, easy to ignore splinter.

It could be argued that Stannis knew about Joffrey. But the same problem remained as when he told Jon Arryn. Stannis making that claim would have looked like it was purely self-serving. That's why he needed Jon or Ned to make the claim for him. Ned was perfect, as he had a reputation for honesty and straightforwardness. But all the proof was in the area of King's Landing. Stannis couldn't have recruited Ned in the North, because he wouldn't have been able to show proof.

If Ned had never spent time with Joffrey, he would likely have been more willing to support him. Particularly if Robert signed a proclamation transferring power to Joffrey. Inertia would have kept Joffrey in charge.

Of course, Joffrey would have still been a train wreck of a king. But without the pressure from Robb Stark in the field, Tywin Lannister could have come to the capital sooner. Joffrey would have had a regent and been less able to make bad decisions.

Stannis might still have caused problems, but he would have needed to be more cautious. And he would have still had the problem of the proof being in King's Landing while he wasn't. Plus of course, the Lannisters would have had more room to eliminate the proof (Robert's bastards). Only the genealogy records would have remained.

In order for Petyr Baelish (Littlefinger) to have gotten the war he sought, he would have needed to work things differently. He might still have done it eventually but not at the moment when Robert died. For that matter, Robert might have lasted longer, as there would have been less immediacy about killing him without Ned's investigation of Joffrey's legitimacy.