Anecdote: At a party, my roommate mentioned casually that people in Greece and the Balkans are white whereas Turkey and the peoples of the Eastern Med are not. I asked him what he thought about the fact that Greeks had been in Anatolia for a long time, and that the Ottoman empire controlled large parts of southeast Europe, and he basically said the dividing line was Islam. So then I asked him about Georgia, and without much hesitation, he said Georgians were not white. I mentioned that they were Christians, and he ended up saying "you have to draw the line somewhere."
Anyway, that got me thinking: how did these peoples conceive of themselves? Was it important to say, Greeks, to label themselves white and Turks as non-white? I'm mostly interested in the 19th century, though I'd like to hear about the 20th century. 18th if it's relevant. I'm not sure how old the concept of whiteness goes back.
People pretty much identified with language and religions in the past, until around the time of colonialism and imperialism. The racial identity of "white" you are describing did not entirely appear in Greece as it did in Western Europe. Before the Ottomans, the Eastern Roman Empire or even the golden ages in Ancient Greece people identified with what their main city was, their language and religions/traditions. They did not identify with the rather recent identity of "white", they were more specific.
Also to explain to you who the modern Turks are Im just going to copy and paste my previous answer. In short many of their ancestors were Greeks and they converted to Islam to not be discriminated against.
A lot of Anatolia became Greek even before Alexander (550-350 B.C.) most of the coastal cities in Anatolia were Greek even long before that and Greek (culture, language) spread through out Anatolia after the end of the Lydian empire. It stayed Greek through Rome/Byzantium. The Swedes, Rus, other slavs and other Orthodox Christians that settled over time eventually became Greek (culture, language etc). Later those Greeks became Turks and the vast majority of Turkey's population were native people in Anatolia who converted to Islam, rather than the actual Turkish tribes and other Islamic peoples who settled there. The reason to convert to Islam is to not have to pay the Jizya (a tax only Non-Muslims called Dhimmi had to pay), they also converted to not be discriminated against or they converted to identify with the victor. Some may have also converted because they liked the culture and faith better than Christianity or Judaism. Also to further answer your question a lot of the Christians (Greek mostly, and others like Armenians who were Orthodox etc.) that were in Anatolia and did not convert to Islam, went to Greece during the 1923 population exchange between Greece and Turkey. You can get a brief account of it in chapter 14 of "The Middle East" by William Ochsenwald & Sydney Nettleton Fisher.