Yea, no. The Roman Empire was de facto a country of two nations - Romans and Greeks. The lingua franca in the eastern parts of the empire had pretty much always been Greek. The New Testament was originally written in Hebrew, Aramic and... Greek. Not Latin.
First, that’s not quite correct. The New Testament is written entirely in Greek, but states explicitly that the language of Jesus and his followers, and most other people in that region, was Aramaic. (See Matt 27:46, John 20:16 and Acts 21:40, among many other places. They convert their first Greek-speakers in Acts 11.) Germanic languages were spoken in parts of the Roman Empire too. And of course, Greece had not been part of the Roman Republic, nor was it within the Roman Empire when it was first established.
If a language counts as Roman because people in the former territory of Rome, although far from the majority, spoke it, in some time period, that would logically apply to languages other than Greek. I don’t want to overstate the point here. The Turks really were foreign conquerors who did not see themselves as Roman. Roman elites had been speaking Greek for a long time. But a lot of what we hear about who do or do not count as the real Romans is just special pleading.
So, to get back on topic, it’s a bit hard to see why the line would be drawn where it is. All the Romance languages have a reasonable claim to be Latin, for example, If any has a better one, it’s the dialect spoken in the City of Rome itself—which is where Charlemagne was crowned Emperor of Rome, in Latin, by the Pontifex Maximus. And, if the “Roman Empire” were reduced to Armenia and the Armenian-speaking Romans decided to change the official language of the Empire to Armenian, it’s hard to see a clear reason why that’s so different from what happened with Greek.