Provavelmente não.
Ele nunca se referiu como Dr Spock, ou como tendo um PhD, em qualquer série da Star Trek (fonte: o segundo dos dois links abaixo). Portanto, não há nada a dizer que ele tenha um PhD, mas também nenhuma confirmação de cânone que ele não tenha.
Você pode encontrar discussões sobre os fãs sobre o assunto aqui e aqui . A partir do primeiro desses links, aqui estão alguns argumentos para Spock não ter um PhD:
We know that Vulcans are not above calling themselves Doctor.
We also know that Spock rejected an offer to attend the Vulcan Science Academy in order to attend Starfleet Academy. I suppose he could've earned a Ph.D. at Starfleet, since he did do at least a little bit of postgraduate work, but most evidence is pointing toward no.
Look, we know Spock is a really smart guy, but that's no excuse for being underqualified. Without a doctorate you're probably not going to get a lot of papers published, so where are you contributing to the scientific community? At the very least Spock made it much harder on himself to gain any credibility for his findings among other scientists.
E:
While you need a large breadth of knowledge in your field to obtain a Ph.D, you also spend a significant portion (2-8 years) of the time working towards it by writing your doctoral thesis, which is incredibly specialized.
Much of what the Enterprise deals with is new phenomena (as well as new lifeforms, new chemicals, new plants, etc). In these cases, a specialist may be of less value than someone with a lot of general knowledge.
In addition, specialists would probably be assigned to ships and locations that would make use of their specialization. Write a thesis on a new theory for warp design and Starfleet would probably have you work in warp design. Write a thesis on an aspect of plant biology, work on in a research station, etc.