The Byzantines called themselves Romans even though they were more Greek than anything. And that is exactly what would happen to the west if it survived. You'd have a bunch of different kinds of people (like Germanics, which were already plentiful in Rome before the empire even collapsed) calling themselves Romans. 'Roman' would no longer be a race, but more of a distant identity that the descendants of Romans, and those of assimilated barbarians and conquered peoples, would take. 

In fact, the Roman Empire collapsed largely due to their inability to assimilate migrants.

To quote wikipedia:
[quote]The Roman Empire began to disintegrate in the early 5th century as Germanic migrations and invasions overwhelmed the capacity of the Empire to assimilate the migrants and fight off the invaders. The Romans were successful in fighting off all invaders, most famously Attila the Hun, though the Empire had assimilated so many Germanic peoples of dubious loyalty to Rome that the Empire started to dismember itself.[/quote]

Rome could not survive while also remaining Roman. Eventually, it would either become Roman only in name, or collapse. Byzantium survived, but it was Roman largely in name. They even spoke Greek mostly.