Well... yes, why not? We are used to see historiographic names (Byzantine, Eastern Roman, Nicean, Latin, etc.) as if it was the the official name of the states and the policies, but it wasn't, once Theodosius divided the Empire, he didn't create two states, the Roman Empire was simply ruled by two guys within their respective zones of authority. The Western half collapsed later and the Eastern remained the sole ruler of the Empire.
"But they were Christians!", so was the unified Empire since Constantine, "they didn't rule from Rome!", so did Constantine, also after him Rome never was the capital again, "they spoke Greek, not Latin!", Greek was already a very influential language since the Republic, but now the Empire ruled a land that was predominantly Greek, so nothing more natural than the government to adapt to it's ruling demographic, or were the Franks not Franks by the time of Charlemagne?