The diary does not prove that he is heir of Slytherin.

The fact the he can talk to and control the Basilisk proves that he is heir of Slytherin.

The diary originally probably contained his notes about how he found out that he was not a muggleborn orphan, but a descendant of that Gaunts and of Slytherin, and how he found clues in the castle about the chamber and the entrance.

In its new form, the diary contains a memory (and a part of his soul), and as we know, memories can be altered. Here Dumbledore and Harry talk about Slughorn's memory.

“As you might have noticed,” said Dumbledore, reseating him­self behind his desk, “that memory has been tampered with.”

“Tampered with?” repeated Harry, sitting back down too.

“Certainly,” said Dumbledore. “Professor Slughorn has meddled with his own recollections.”

“But why would he do that?”

“Because, I think, he is ashamed of what he remembers,” said Dumbledore. “He has tried to rework the memory to show himself in a better light, obliterating those parts which he does not wish me to see. It is, as you will have noticed, very crudely done, and that is all to the good, for it shows that the true memory is still there be­neath the alterations. (HbP)

The diary is of great personal importance to Riddle because it about the time when he discovered that he was a heir of the founders, but it doesn't prove anything to anybody.

You are probably referring to this scene of the same book between Harry and Dumbledore

“And they could be anything?” said Harry. “They could be old tin cans or, I dunno, empty potion bottles. …”

“You are thinking of Portkeys, Harry, which must be ordinary objects, easy to overlook. But would Lord Voldemort use tin cans or old potion bottles to guard his own precious soul? You are for­getting what I have showed you. Lord Voldemort liked to collect trophies, and he preferred objects with a powerful magical history. His pride, his belief in his own superiority, his determination to carve for himself a startling place in magical history; these things suggest to me that Voldemort would have chosen his Horcruxes with some care, favoring objects worthy of the honor.”

“The diary wasn’t that special.”

“The diary, as you have said yourself, was proof that he was the Heir of Slytherin; I am sure that Voldemort considered it of stu­pendous importance.” (HbP)

I didn't find any indication that Harry did consider the diary to be proof of Riddle's heritage, although Dumbledore here claims that Harry did.

It's probably just a shortcut for Rowling because she didn't want to elaborate why the diary was of great importance to Riddle, one of the "objects worthy of the honor".

The diary is not that important or special to the general public, it is important to Riddle, probably more important than some historical objects, especially those of the other founders. The diary is about himself, the most important person in his not so humble opinion.

Editar

To add a little more context to your quote:

(Dumbledore:) “Well, it worked as a Horcrux is supposed to work — in other words, the fragment of soul concealed inside it was kept safe and had undoubtedly played its part in preventing the death of its owner. But there could be no doubt that Riddle really wanted that diary read, wanted the piece of his soul to inhabit or possess some­body else, so that Slytherin’s monster would be unleashed again.”

“Well, he didn’t want his hard work to be wasted,” said Harry. “He wanted people to know he was Slytherin’s heir, because he couldn’t take credit at the time.” (HbP)

Here Dumbledore and Harry are guessing about Riddle's motives, and I think they are wrong.

The important point about the diary is that it is a Horcrux, it allows him to survive if his body is destroyed. Riddle really wanted that diary read, so that it could suck the life energy out of someone, so that he could come back to life. That is much more important than whether or not the Basilisk kills some people.

Riddle also wouldn't want the diary read unless his body was destroyed, because then there would be suddenly two Riddles, the main one and the one created from the dairy, and they would not want to share power, not even with another version of themselves.

Voldemort is considered a Slytherin

“Better Hufflepuff than Slytherin,” said Hagrid darkly. “There’s not a single witch or wizard who went bad who wasn’t in Slytherin. You-Know-Who was one.” (PS)

Although it is unclear whether he is considered the Heir of Slytherin.

But he speaks Parseltongue and I'm sure at least his followers believe him when he says he is the Heir of Slytherin. Dumbledore also knows that it is true, and I doubt someone would be foolish enough to claim otherwise in public. Voldemort is not the type to come with proof to argue and convince someone, he is the type to kill dissenters.

Also, Voldemort is not the type to kill someone for credit, he does it for fun and for power.

In short, Voldemort has no need to prove that he is the Heir of Slytherin, and the diary is not useful if he wants to prove it.

It’d open the Chamber to prove him as heir.

Tom Riddle planned that his diary, after being made a Horcrux, would be used to reopen the Chamber of Secrets someday. That would then show that he’s the heir, since the legend of the Chamber says that only the Heir of Slytherin can open it.

“The story goes that Slytherin had built a hidden chamber in the castle, of which the other founders knew nothing.

‘Slytherin, according to the legend, sealed the Chamber of Secrets so that none would be able to open it until his own true heir arrived at the school. The heir alone would be able to unseal the Chamber of Secrets, unleash the horror within, and use it to purge the school of all who were unworthy to study magic.”
_{^{- Harry Potter and the Chamber of Secrets, Chapter 9 (The Writing on the Wall)}}

Through the diary, he’d intended to lead another student into opening the Chamber and release the basilisk. Though he never said explicitly how he’d ensure he was known as the true heir, presumably he’d claim credit for it at that point, either by having the student state it for him or by appearing from the diary himself.

“I knew it wouldn’t be safe to open the Chamber again while I was still at school. But I wasn’t going to waste those long years I’d spent searching for it. I decided to leave behind a diary, preserving my sixteen-year-old self in its pages, so that one day, with luck, I would be able to lead another in my footsteps, and finish Salazar Slytherin’s noble work.”
_{^{- Harry Potter and the Chamber of Secrets, Chapter 17 (The Heir of Slytherin)}}

Both Dumbledore and Harry say that Tom wanted to reopen the Chamber and release the basilisk so people would know he was Slytherin’s heir because he couldn’t take credit for it while he was still attending Hogwarts.

“But there could be no doubt that Riddle really wanted that diary read, wanted the piece of his soul to inhabit or possess somebody else, so that Slytherin’s monster would be unleashed again.’

‘Well, he didn’t want his hard work to be wasted,’ said Harry. ‘He wanted people to know he was Slytherin’s heir, because he couldn’t take credit at the time.’

‘Quite correct,’ said Dumbledore, nodding.”
_{^{- Harry Potter and the Half-Blood Prince, Chapter 23 (Horcruxes)}}

Therefore, he was likely going to use the diary to prove he was the heir by opening the Chamber of Secrets through it and making it clear he was the one who opened it, since the legend surrounding it says only the heir can open the Chamber.