yes it did,Christianity probably want be as wide spread if it went for Constantine,who was about to fight a war,he was said to have seen a image of Jesus,he commanded his warriors to paint crosses on their shields(he won the war).and he stopped the calcification of Christians and made Christianity the main religion of Rome.
Rome did play a major role in the spread of Christianity. However, until Constantine (4Th century AD) Rome was a heavily persecuted religion. The truth of the matter is yes Constantine did make Christianity the official religion of Rome, however before he did that Christianity was already fairly widespread. It is tough to think what it would have become without Constantine's legalization of it, however my guess is it still would have been fairly strong. It had already been spread all across north Africa and much of modern-day turkey and was a force in Rome before Constantine. Therefore, I don't think we can say that Rome is solely responsible for the spread of Christianity.