Hey CJPrker,

Your question rests on the assumption that, at some point, the Eastern half of the Roman Empire had actually been culturally Roman or Latin. The thing is that it never really was.

Most civilizations in the Western half of the empire were slowly assimilated in to the Roman empire after often drawn-out and devestating wars. The most extreme example of this assimilation policy was the fate of Carthage, which was almost completely destroyed. Their empire was, culturally speaking, the only worthy rival of Rome in the West. The Carthies are an exception though: most of the Western civilizations were tribal, lightly populated and usually not very urbanized. The Romans built many colony-settlements in the West and the Gallic and Iberic elites were soon thoroughly 'Romanized'. Through these elites, Latin/Roman culture became dominant in the West.

In contrast to the Western half of the Roman Empire, the East had already developed many different and complex urbane civilizations by the time they were subjugated by Roman armies. A key difference is that, in contrast to the West half of the empire, the East already had a common culture: this was the Greek/Hellenistic culture inherited from Alexander and his successors. Where in the West, Carthage was the only city which could rival Rome in terms of size and influence, the East contained many large and culturally accomplished cities like Alexendria, Antioch, Pergamom and many more.

Next to that, the East was swiftly and (relatively) non-violently conquered by the Romans. Ptolemaic Egypt, for example, became 'officialy' Roman after the battle of Actium, a naval battle on the Greek coast. Pergamom became a Roman province after it's king gave his realm to Rome as an inheritance. The Seleucid Empire was slowly devoured by the Roman and Persian Empires over the course of the 2nd and 1st century BC. The point I'm making here is that Roman expansion in the East did not come with the same scale of warfare and destruction as Roman expansion in the West.

So to wrap it up, the Byzantine Empire had never been Roman or Latin in culture at all, it had been culturally Greek since Alexanders age and remained that way until the rise of the Arab Empire.. Latin was never a lingua france in the East, it was at best used cermonially. If you're interested in the transition from Eastern Roman to Byzantine Empire, I can redirect you to an answer to a similar question I gave a while ago