>spoke Greek
They spoke Latin as the official language until 610, and spoke Greek after. A change in language does not mean an abandonment of being Roman.
>Also, the city of Rome itself not being part of the Empire is a large factor in the objective answer being 'no.'
Was the Eastern Empire under Diocletian not Roman? If London became an independent city state, would England cease to be England? Would the English identity vanish? Rome was not even the capital for many periods of Roman history, in fact a few emperors only visited Rome once, if at all.
What do you even mean by an "objective answer"? I see "objective" thrown around a lot on Veeky Forums. Do you mean it in the philosophical sense? If so, you need to post some axioms.
>were Orthodox Christian
Not for the full history, they were pagan, then Christian -- the Western Side was pagan, and then Christian
>the empire was a Monarchy
The rules of succession were not clearly defined from the start, and throughout Rome's history, the methods of imperial succession changed. Each emperor was essentially a monarch however. Do you argue that because the Senate was abolished, it became a monarchy? You can have monarchs with a senate -- just look at the majority of Roman Empire history.
>So, culturally, they weren't Roman, at all
"Roman Culture" as with many cultures, changes through time. The art under the late emperors is very different to the art under Augustus.
It began as a split of power in the Roman state; it was ruled by Roman Emperors in unbroken succession until 1204 by which various claimants emerged.