I suppose the distinctive Greek feature was the Orthodox church.Well, maybe getting a bit OT by now but it was never a Greek empire. Of course it's a matter of what parameters one applies but culture alone won't cut it.
Quite apart from which its official language remained Latin until well into the 7th century when Herakleios (who DDD mentions) began "grecianizing" it.
There had been no independent entity "Greece" since heck knows when, if indeed there ever had been one at all. So no Greek empire could have arisen (Even under Alexander it was the Macedon empire).
Constantine didn't found a Greek empire, he took power over all of Rome and then made administrative changes, one of them consisting of building a new residence at Byzantium (a town). His place of birth had nothing to do with anything that would have made him un-Roman, certainly as little as Hadrian's Spanish descent would have made his rule Spanish.
That Greek was spoken thruout the Roman empire and its culture heavily influenced any Roman (and not just those) made Rome as little Greek as the US is English. That goes for what became known as Eastern Rome (with the "Western" decline) as well.
Indeed Roman armies re-captured a lot of the land that the "barbarians" had vested from what is now the Italian peninsula. That they came from the East got nobody calling them Greek armies.