I think the Byzantine Empire was best described as Greco-Roman. Greek was their main language and their culture was also very Greek, but they called themselves Romans and in fact were simply the eastern half of the Roman Empire that kept on ticking while the western half fell over, and so they had a profoundly Roman culture as well. Additionally, their proximity to the "east" (ie. Persia) caused some appropriation of eastern culture, their heavy use of horse archers and cataphracts good examples in the military sense.

What it was to be Roman was, the third century or so, pretty closely tied with Greek culture. Roman emperors such as Julian or Valens and their entire officer class wished to emulate Greek and Hellenistic heroes, and the Roman intellectual class was fluent in Greek. Greek culture infiltrated that of Rome with a vengeance from the 2nd century AD, though Roman fascination with Greek thinking and history dated to the 2nd century BC.

As the Roman empire matured it might have become accurate to describe West or East as fundamentally Greco-Roman, but while Western Rome was profoundly influenced by Germanic cultures, Eastern Rome stayed more Greco-Roman. And the Byzantine Empire was not only the successor of the Empire, it was the Empire. Again, they considered themselves the Romans, not the Byzantines. That's just a term that got stuck to them for ease of reference.

Note on sources:

I don't have sources for you right now, I typed this from my phone while lying in bed. Sorry...