I don't buy the black-and-white doom scenario that Muslims will overrun Europe and stamp out Christianity. In fact, they probably won't even expand beyond Greece. Bulgaria is a powerful neighbouring state that pretty much stops their northward expansion, Anatolia will probably host a set of rebellious warlords and strategoi who need to be mopped up, and all of the conquests need to be digested by the already very-overstretched Umayyad caliphate (mind you that abandoning Spain as a whole was genuinely considered by them). The expansion of the caliphate is almost certainly going to come to a complete, screeching halt after Greece, with no further territory taken.
The Romans/Byzantines are absolutely boned but they aren't dead. Much of mainland Italy, along with Sicily, Sardinia, Corsica, Malta, the Balearic Islands, and the Dalmatian coast, are still under imperial control. Most likely,
Basil Onomagoulos or another guy declares himself emperor in Sicily or mainland Italy, depending on the exact timing of Constantinople's fall. With some respite from the Arab threat, the Romans (in essence a second Western Roman Empire) will likely have sufficient breathing room to consolidate their possessions and rebuild their state. The capital would probably be moved back to Rome.
The Greeks are
not going to Arabize. If anything, they'd likely be a second Persia; one that maintains its own language, culture and identity even after adopting Islam. The Romans couldn't assimilate the Greeks and I doubt the Arabs will succeed either. Greek had, after all, being the
lingua franca of the Eastern Mediterranean since the time of Alexander the Great. An interesting knock-on effect of this is that African Romance and Coptic might also survive, considering that the whole Greek-speaking world is under Arab dominance (the same thing happening to the Iranian-speaking world was thought to be a major factor in allowing Persian to sustain itself); Arabization might actually be much more restricted ITTL.
Constantinople would almost certainly become the Umayyad capital, if they can hold the city. Making it their capital was a long-term goal of the Arabs. Whether they'll keep it is another question; perhaps the Greek part of the caliphate might split off in much of the same way that Persia did, going its own way. It's a pretty interesting scenario to think about.