Is that really the case? Seems more like wishful thinking to me. Christianity was spread globally through missionaries. liberal enlightenment - not so much.
Colonialism? Or else why does the UN extend throughout the globe?
The founders of Christianity were Jews. Early churches were in Ethiopia, Israel, Turkey, Egypt, Syria, and, yes, Rome and Greece. It was a very multi-cultural movement.
(edit: strictly speaking, the Eastern Roman Empire didn't exist yet by that name. It was the eastern part of the Roman Empire)
Whilst the anatolians had sort of a partial continuity into Islam by the middle of the millennium, a process that completed with 19th-20th century genocides of ottoman christians. (Off-topic: I'm reading into comparative genetic reseach about anatolian/near-east peoples, of which I'm a member, which suggests that there is a continuous people since the paleolithic here, that is divided differently at different times in the history because of culture replacements of imperial origins, i.e. Parthian, Armenian, Greek, Roman, Muslim and Turkish empires. But especially the Turkish-speaking, the Armenian-speaking and the Kurdish-speaking peoples seem to be genetically very close to suggest common origin, whereas Anatolian turkish-speakers are genetically closer to brits than central-asians.)
Why wasn't Christianity more popular among the Jewish people? The Bible has some theological answers to that question. I wouldn't consider a Greek heritage to Christianity to be a main cause of that though. Jesus and the apostles were all Jews, after all.
So many shifts and things have happened since christianity enters the european continent that it can't hardly be the foundation of the common european identity. Just look at who started the EU in the 50's and led us to believe in an european dream.
Point given: it's an argument used to antagonize proponents (nationalist leaders and `muslims`(whatever that group really is)).