Not only did the Byzantine Empire refer to itself as the Roman Empire and the Citizes to themselves as "Roman" but Greece never held Constantinople or anything past Macedonia.
I am not denying that there were Greeks in the eastern Roman Empire but it was definetely not just Greeks, and the citizens never called themselves Greek.
(This is not me hating Greece, I am Greek and I just find it confusing when people refer to the Byzantine Empire as "our ancestors")
Byzantium was a Greek city to begin with (there were Greeks all the way around the Black Sea and Anatolia and further into Asia at this time) when it was decided by Constantine to move the seat of the Roman Empire there. It was, however, still in a predominantly Greek area and within a couple centuries Greek even became the official language of the empire and it took on a distinctly Greek flavor. They did continue to call themselves Romans and they were legitimately the continuation of the Roman Empire, to this day the Turks refer to Greeks as Rum (Romans), but it became a thoroughly Greek empire distinct from Rome and it's former Italian foundation.