Depends on what you mean by 'turning Greek.' The Eastern half included territories that were either Hellenophone by majority or ruled by Hellenic dynasties, as well as having the existing Coptic, Aramaic, Hebrew, and Anatolian (to a degree) and one group of Gauls around. The Jews were the significant exception to the otherwise straightforward rule that the Roman Empire did not change a very great deal of the Hellenic cultures beyond deposing their rulers, given the wave of a second (perhaps third) Hellenization that affected Imperial Rome as much as it did the Republic before it.

The pagan Empire made no great deal about administrative details nor much of an attempt to 'Romanize' or 'Latinize' subject peoples in the East, where civilization and cities were far more ancient than Rome itself....bar the case of the Jews who fought three big wars and lost all of them with the Roman Empire. So in that sense the East was always an Empire of the Greeks.

The change in medieval Rome from what was a clear survival of the Eastern half of the empire to what is commonly called 'Byzantium' would be the era of Heraclius, where the ruinous victory and more ruinous defeats against Sassanians and the first Islamic armies truncated away the areas that spoke non-Hellenic languages, as well as non-Chalcedonian Orthodoxy as well. So the medieval transformation was the result of the Islamic conquest of North Africa, Palestine, Egypt, and Syria. It erased the non-Greek Orthodox communities and left a relatively more culturally monolithic autocracy that probably did much to strengthen the cohesion of the Byzantine Empire than it did to weaken it.

So it depends on which version of 'Greek'. Are we talking the heirs of the Ptolemies and Seleucids of pagan Rome or the proto-Russia of medieval Byzantine culture?