What If?

Am reading A Short History of Byzantium by the rather pompously named John Julius Norwich.
An elegant writer, Norwich, after describing Byzantium and the magical resonance of that place-name, writes:
Finally, the man: Constantine I, Emperor of Rome. No ruler in all history has ever more fully merited his title of 'the Great'; for within the short space of some fifteen years he took two decisions, either of which alone would have changed the future of the civilised world.

The first was to adopt Christianity as the official religion of the Roman empire. The second was to transfer the capital of that Empire from Rome to the new city he was building on the site of old Byzantium... Together, these two decisions and their consequences have given him a serious claim to be considered--excepting only Jesus Christ, the Buddha, and the Prophet Mohammed--the most influential man in all history...

I don't remember if I've learnt that in school, but reading it now and thinking of the 'what ifs' gives me goosebumps. If Constantine had not adopted Christianity, that Jewish reform movement would probably have faded away, like tens of other religions that glowed for a brief while before either fading away or being stubbed out .
So, would the whole world have been Moslem? Imagine domes, cupolas and the muezzin's call in London, Paris and Rome. And also, no Crusades, no overdressed Popes and even the Iraq war?