So to answer your question, it's important to first point out that there's a reason the term "Greco-Roman" is so prevalent when talking about the history of the Mediterranean. Further complicating matters somewhat can be that at many times, culture was seen as exclusive to the Greco-Roman world. There was Greek and Roman culture and everyone else was barbarians.

Even before Greece was integrated into the growing domain of the Rome, there was an effort to associate Roman culture with that of the Greeks. Most visibly in religious practices where Zeus became Jupiter, Ares became Mars etc. Alexander the Great was romanticized to the point where he was the idol of practically every Roman man that fancied themselves a leader.

The influence of Greek culture only grew when Greece came under the control of Rome. Greek slaves began to come over in droves due to their skill in a variety of different areas. Educated slaves from Greece became educators and taught the Roman youth. Greek craftsmen and artisans built the buildings and sculptures that showed off Rome's wealth and power to all. They also helped manage the growing bureaucracy of the Empire. Linguistically, Greek was one of the two lingua franca of the Roman world with most of the elite at least partially bilingual.

So how much of Roman "culture" was consciously retained by the Eastern Roman Empire in the centuries following the end of Roman rule in the West is hard to say since they were invariably intertwined. Justin I, hailing from Illyria was likely the last Emperor that spoke Latin as his first language. Although the influence of the Latin language began to fade, there was never a point where they saw themselves as distinct from the Empire of Augustus. Culturally or otherwise. Something that is facially obvious if you consider how many Byzantine Emperors were named Constantine after "the first Christian Emperor." Their view of religious hegemony as belonging exclusively to the State is something that evolved as part of Greco-Roman culture throughout the period of unified Imperial rule.

Much of the institutions we associate as "Roman" continued to survive in the East. The Senate established by Constantine in the Eastern Empire continue to exist (at least on paper) until the 14th Century with the title of Senator enduring even after that. The title of Sebastos, the literal translation of Augustus likewise continued to be used as were others such as Kaisar (Caesar) and Anthypatos (proconsul). How long such titles were in use and the degree of importance they carried is sometimes difficult to determine as the system of titles the Empire used is rather confusing and complicated to the point that "Byzantine" is a term to this day one can use to describe something unnecessarily complicated.

That is not to say that the East didn't gradually develop its own unique culture. Far from it. For example the Byzantine government became a virtual theocracy with the growing influence of the Orthodox Church. The Empire was literally ruled by God through the Emperor with it being the earthly equivalent to the Kingdom of Heaven.

But critically import to your question is the negative impact of Byzantine religiosity on understanding certain parts of their culture. This is because of two periods of rabid iconoclasm where many of the beautiful mosaics we associate with the Byzantine period were destroyed as idolatry. So much so that the most famous surviving examples such as the Mosaic of Justinian I only survived because it was located outside of territory under Byzantine control.

This religiosity also had apparent influences on their literature as out of the two-three thousand surviving examples of Byzantine literature we have, only around 300 could be described as secular.

Finally, to put a bow on my interpretation of the question is that the Byzantines never stopped calling themselves Romans. They never stopped calling their Empire "The Roman Empire." The various forces that came into conflict with them from the East never stopped calling them Romans. The title of "Emperor" was never not interpreted as the pinnacle of Medieval hegemonic power with even the Ottomans claiming the title for themselves.

To say they saw themselves as culturally Roman would be very much accurate. Would their culture be the same as that which we associate with Caesar, Pompey, Cato and Cicero? No. But neither would that of the Empire under Domitian. I struggle to think of any aspects of Byzantine culture could be accurately described as "continuations of pre Roman Greek culture" because Roman culture was so heavily influenced by Greek culture.