The official language did not change to Greek until the empire nearly collapsed as a result of the plague, Persian wars and the Arab invasion.
The state that emerged from this was were different to the Roman Empire rules by Justinian in many ways.
> after the Fall of Rome was
I would disagree that was the case in the 500s, the Fall of Rome was mostly seamless and most actual changes occurred years later.
so it may be true that the Goths in Italy were Romanized (as had repeatedly happened with conquerors, interestingly), the empire that remains after the Fall of Rome was, in total, different in character, culture, language and territory.