Culturally, latin to greek might not be what you're looking for. The Eastern Roman empire did shift to greek linguistically - but that was because much of their territory were native Greeks. Earlier on, there were some others - Latin in the western Balkans, Africa, and Italy, Coptic in Egypt, Syriac on the east coast. But Greek had long been the lingua franca of the Eastern Mediterranean.

But is the language what makes the culture? Were the eastern Romans in 200 AD, speaking Greek, Syriac, Coptic, etc - not Romans? They considered themselves Romans. They were part of the Roman empire. They just didn't all speak Latin.

Well, it's the same thing in later times. The Eastern Roman Empire (or Byzantium) considered themselves to be Roman - all the way up until the fall of Constantinople in 1453.

There's some details in this earlier answer - https://www.reddit.com/r/AskHistorians/comments/7tnjr9/how_much_and_how_long_did_the_roman_culture/dte98p5/