Christianity did come from Rome. If you read the biblical book of Acts, it describes the origins of the early Christian church coming from some where near the Mediterranean Sea. Rome is a logical place for this, as Jesus was born in the time of the Roman Empire and Rome is a cultural center.
Christianity began a Jewish sect in Levant, a place in the Eastern Mediterranean. It did spread from there and Rome would eventually become Christian in many ways, but Levant was the origin, not Rome. In fact the Roman Empire historically even discriminated against Christians, to the point of executing them or setting them up to die in arenas.