I'm sure many Greeks did not convert to Christianity and I'm sure many converted to Islam but as far as I know Greek identity was firmly Christian. I understand that a nation confined to a territory may well convert en mass (particularly if they have are under a unified political system) but why did the Greeks of Turkey, Egypt and elsewhere uniformly convert to Christianity and then resist conversion to Islam without political coordination? I assume the answer is that many did convert to Islam and ceased to be a part of the Greek diaspora, but if that is the case then why did those that did convert uniformly fail to maintain a Greek culture as Muslims?
When the Greeks converted to Christianity, this was because they lived in the Roman Empire, which in the 4th century became a Christian-run state (obviously some had converted pre-Constantine, but it was a minority) with Christian institutions encouraging Christianity, and in 380 actually became officially Christian. Not to say that they were forced to do so, but it became more and more popular/influential/powerful until basically all Roman citizens (including all Greeks). By this time, Rome was seen as, essentially, God's Empire by Greeks (but perhaps not by Armenian Christians, Arab Christians, Persian Christians, etc!). So Christianity and Greek identity became intertwined. Now, when the Arabs conquered areas like Egypt and Syria, they did not initially do a lot to convert the locals. So many Greeks continued to be Christians, but more importantly, they knew that in Greece itself, there was a Greek-speaking Emperor ruling over a Greek Christian populace. In addition, the holy texts of Christianity were (at least the second half) written in Greek. So, Christianity was very much associated with Greek culture. (I think it's probably important to note though, that they weren't referring to Greece as Greece, but as the Roman Empire, and its people as the Romans.)
Now, over time (especially after the Abbasid revolt), the Caliphate became more Arab-ised, and this was mixed in with Islamification. The Qu'ran was in Arabic, the children of Greek/Arab couples would generally be considered Arabic (as by religious law, the father would be a Muslim, so probably an Arab), etc. This period saw Syrians, Mesopotamians, Egyptians, and Berbers essentially become culturally Arab, to the extent that we (and they) refer to them as Arabs today. So it's not just Greek identity was Christian, but that Muslim identity was Arab. This was further encouraged by the Ottoman Empire, whose Millet system classified people by religious group. Christians were either "Rum" (ie Romans, ie Greeks), headed by the Greek Patriarch, or Armenians (later other kinds of Christian were recognised). Thus, to be a Greek Christian was to be part of a social identity run by Greeks, whose courts were in Greek. Whereas to be a Muslim meant to be in the majority group, to speak and interact with Arabs or Turks, to be subject to Sharia law, and so forth. So those who converted to Islam lost their Greek identity.