In every way, your statement is equally correct with Christianity and Christian substituted for Islam and Muslim.
Heck, the statement is pretty much correct with "multiculturalism" and "Canadian" substituted for Islam and Muslim, or "democracy" and "American". It's basically just a tautology; cultures want to spread.
The trick is being successful. I've seen nothing to indicate that Islam spreads "successfully". A little education and money seems to mellow out Muslims pretty quickly.