This really only supplements Dennis De Bernady's answer. The Eastern half of the Roman Empire, which we call Byzantium after the former name of its capital city, had previously been part of the Greek/Macedonian ruled Empire of Alexander the Great, and the successor states into which it split after his death founded by his generals e.g. Ptolemy in Egypt. Hence Greek was the language of the rulers and became the 'lingua franca' of trade and cultural exchange. When the Romans took over the region, as government and commerce were already mostly conducted in Greek and this seemed to work, and educated Romans already tended to know Greek, there was no reason to change it.
When I have visited former parts of the East Roman Empire like Greece, Turkey and Egypt where ancient inscriptions remain from the Roman period they are very, very rarely in Latin, but almost always either in Greek or a local language.
The New Testament was written in Greek as the main literary language of the region even though it was then part of the Roman Empire, Jesus preached in Aramaic and the scriptures he quoted were in Hebrew.
When the Western Latin-speaking half of the Roman Empire fell to the barbarians in the fifth century, and the attempt at partial reconquest under Emperor Justinian in the Sixth Century did not last, most of the Latin speaking parts of the Empire were gone, and the Greek speaking parts continued speaking Greek, so Latin fell out of use.
This was not 100%; speaking of Latin in the formerly barbarian provinces of the northern Balkans had been encouraged and survives today in the Latin-derived language Romanian (whose name is related to Roman) and until the nineteenth century in the now extinct Dalmatian language.
Also, when Roman law was codified under Justinian in the sixth century it was done (like the earlier laws on which it was based) in Latin, producing the 'Digest' of laws which is the main form in which Roman law was studied in Europe thereafter.