In the second half of the sixth century or first part of the seventh century much of the Balkans was overrun with Slavs (including some parts of Greece which were later re-hellenized). Although much of the territory was again under the control of Constantinople from time to time, that must have tipped the scales heavily in favor of the Greek language as opposed to Latin. Justinian's law code was in Latin but already a lot of the new laws were being issued in Greek.
kabumpo: "Justinian was in Byzantium, the eastern part of the empire that lasted longer."
Verginius Rufus: "Justinian was from near Skopje in the present-day Republic of Macedonia--one of the last emperors who spoke Latin as his first tongue..."
Just to be clear, here is the old Roman Empire at its peak, 2nd Century AD:
And here is the new Eastern Roman Empire (possibly exaggerated?), at its peak under Justian, 6th century AD:
The Eastern Empire fell in 1453, and now, as the song says:
;-)