Latin is the official language of the Catholic Church, not all (or even most) churches and for much of its history Latin was arguably below Greek in importance. Greek, or more formally Koine Greek comprised a sort of lingua franca in the east at this time, spoken in Greece itself, the Greek colonies in Asia minor, the latter foundations by Alexander stretching all the way to the borders of modern Pakistan and in the great urban centres of Antioch and Alexandria. Further, because of the proliferation of Greek slaves throughout the Empire there were substantial Greek speaking populations in many other Roman cities without a native Greek population. When Christianity really began to distance itself from Judaism many of the earliest converts were Greek speakers (particularly among slaves) and everyone who had a good education understood some of it. The point essentially is that Greek was the language that everyone spoke and was the best way for communicating a message of this type to people who would be receptive.

Just because Jesus spoke Aramaic does not mean that his disciples, the ones that spread the message outside Judea solely spoke Aramaic either. Paul for instance certainly spoke Greek and preached in it while he was in Athens. Later on the Church became ever more hellinised because many of the early church fathers who codified the major doctrinal points of Christianity such as Origen and Irenaeus were Greek speakers. Personally I do not think comparisons between Arabic being the language of Islam and Latin being the language of Christianity are at all apt. If anything the first language of Christianity is Greek.

As to the idea that the church could "require" believers do anything at all we have to remember that the Church was scattered, persecuted and weak. The great debate about the nature of Christ was not finally settled until 325 at the council of Nicaea and there were numerous doctrinal disputes and heresies for centuries afterward. As for why Latin became the language of the church in the west I would guess that it was because that was where Latin was spoken, however I cannot give a detailed answer so I will leave that to someone else.