Well, no, that's just silly - but the Romans in Constantinople were more than happy to marry in to the new Ottoman dynasty. They were just so abandoned by Europe at that point, so left to fend for themselves, and the Ottomans were looking for legitimacy to their rule so they figured it was a good trade: the Roman lines could live on in Istanbul and the Ottomans got to claim that they had ascended to their rightful place as the continuation of the Roman Empire.
The Greeks will occasionally bitch about how Byzantinos was originally theirs, but that's a difficult claim to sort through - the Greek "identity" didn't really have a clear meaning until Thermopylae and we're talking about a city that was then part of Persia anyway. It's tenuous.
It's certainly not as clear cut as the aboriginal claims in Australia.