I don't think the genealogy is that convenient - the Eastern Roman Empire arguably has a stronger claim to being the successors of "The Greeks" that are often claimed to be the root of "Western" civilization than any place in Western Europe. Constantinople became the capital of the entire Roman Empire by 324 and the importance of the West (and Rome itself) for the empire dwindled already hundreds of years before the official "fall" of the Western Empire (which is arguable, among other things, because the "Gothic" king Theodoric received the official Western imperial regalia from the Eastern Emperor, after Odoaker had sent them to Constaninople when he deposed Romulus Augustulus). The Gallic provinces even spent some time as an independent Empire during the Crisis of the Third Century.

In the end, the modern name of "Byzantine" seems to be misplaced - these people were the Romans.