Assume that the Roman Empire never falls (something like the West never loses Africa, allowing it to gradually recover and re-unify), and maintains borders comparable to its classic borders. Basically, as long as they control the entire Mediterranean coastline, we're cool.
Could the institution we know as the Medieval University, with all (or most) of its independence, have arisen in such a civilization? On the one hand, the idea of Church and State as separate is already there, after Ambrose. Plus, there were a variety of ways in which the Medieval Universities began, some organized by professors, some by students, others by local leaders. On the other hand, the government might be as skittish as ever around the idea of independent and exclusive groups, like a university would represent.
Other food for thought is that the rest of Europe is likely developing along lines not too dissimilar to history; Europe north of the Empire is growing more prosperous, more populated, and more civilized. There will be an environment in which higher learning is valued. Might this be suitable for the development of universities, even if the Empire itself is not? Or might universities within the Empire be seen as a way to Romanize those beyond the borders, with the students of the elite traveling to the Empire to be inculcated in their civilization?
Go nuts!