Basilisk1,
I too enjoy this conversation.
It's been intellectually stimulating, and a pleasure to the mind.
For your example to be more apt, let's say that the thirteen colonies did not revolt in 1776. Now, the Americans would still consider themselves British and loyal to crown and parliament.
In 1805, Napoleon somehow was able to land his grand armee into Britain and occupied the British Isles, and the British King and Parliament fled to the North American continent. They set up shop in, say, Boston or New York, and they failed to reconquer the home islands.
Every institution of the old country was adapted wholesale into the new continent, although it would be substantially modified by the new environment.
Now, the culture of French occupied Britain and the British North America would substantially diverge in time that in about three hundred and four hundred years, the language and culture would be totally different. The British would island would speak a modified kind of French, while the North American would speak some kind of English and cling to the traditions and pretensions of Britain even if Britain isn't in their hands.
Now which of the two lands could more plausibly claim to be the continuation of the British tradition?
The reason why the United States of today is different from the UK. They consciously and deliberately separated themselves from the British and clearly and loudly shouted those distinctions to the world.
The parallel would be if the provinces of Greece, Macedonia, Thrace, Asia Minor, Syria, Egypt, and Judea consciously declared their intention to secede, and call themselves the Byzantine Empire, and state that they would categorically deny any involvement with the legitimate government in Rome.
However, I don't deny that Constantinople in 1453 is far far different from Rome in 476.
Culturally, the Classical Rome and Byzantium are very different, and if civilization is defined purely on basis of culture, then Byzantium and the Rome of Augustus is different civilizations.
But a civilization is more than culture.
A Greek living in Tarentum in 45 BC could say that he is under Roman civilization.
A Greek living in Athens in AD 120 is a Greek, yet also part of the Roman civilization.
A Samnite living in Capua in AD 70 could say he is under Roman civilization.
A citizen of the empire in the time of Diocletian would be part of the Roman civilization, whether that citizen speaks Latin, Greek or Aramaic.
Simply put, Roman civilization and being Roman simply means being a Roman citizen inside the boundaries of the Roman Empire.
The state that survived the fall of the West, was, as you stated, Roman, and called the Roman Empire. The cultural mix and basis of the empire was radically altered, but the basis of the Roman civilization was not an exclusive culture, otherwise people like the Sabines and Samnites and Spaniards and Greeks and Jews could not become Roman citizens hence members of the Roman civilization itself.
So the Byzantine Empire is in fact the Roman Empire and the civilization could properly be called the Roman civilization, even if much of the civilization took from Classical Greece than from Classical Rome.
The fact is, it is possible to be Greek speaking and part of the Roman Empire and part of Roman civilization during the time of Scipio, Caesar, Trajan, Diocletian, Justinian, Heraclius, Basil, Alexander Comnenius, or even Constantine XI.
Roman civilization is not exclusive to the Latins and Latin speaking people, nor does it exclude anything Greek.
Roman and Greek are not mutually exclusive concepts during this time.