The Roman Empire fell officially in the year 476 AD, it included the city of Rome as its capital (for the most part), it spoke Latin as its first language.
The Byzantine Empire isn't the Roman Empire because of those reasons. It never was the Roman Empire, even as the Eastern Empire, because, it was independent from Rome. So, Byzantium is not Rome, has not right to be called Rome and shouldn't be considered Rome.
Rome as a city had become divorsed from the idea of Rome the state a long time ago, very few of the western emperors ruled out of Rome and the eastern emperors kept on speaking latin well into the 8th century, frankly to say Rome fell with the west ignores the fact Rome had changed so much in this timeframe that in effect the core region had shifted to the east.