How successfully would Islam be able to spread without a state apparatus enforcing it and carrying out conversion by the sword, though? Regardless of the refined philosophies that the Islamic golden age later produced, early Islam relied heavily on force. Areas were conquered, and the conquered inhabitants were converted as a result. Without a strong enough polity around to do the conquering, the only other means of spreading Islam in that early stage is demographics. That is: Arabs migrating north. Not conquering, but moving into existing Empires (the ERE and Persia) like the Germanic peoples moving into the Roman Empire-- and taking their religion with them.
I'm not at all sure that would be effective. The Germanic peoples did not suddenly convert Rome to worship their gods, after all. If anything, it was the other way around. Ceteris paribus, the dominant religious and cultural norms win out. The victory of Christianity over traditional Roman beliefs had more to do with the totally different approaches they had: Roman religion was heterodox, inclusive, didn't have a central authority structure or a canon of holy books.... and Christianity was orthodox, exclusive, did form a central structure of authority (a church, a clearly defined clergy bound to that church) and did have a defined canon. Things between those two forms of religion were not equal. Islam would have the same advantages Christianity had against pagan practices, but it wouldn't have those advantages over Christianity, nor over Zoroastrianism.
Unless Islam develops a tradition of travelling philosophers who instruct people in their wise ways, I don't see it spreading very successfully without state backing. Of course, Islam did develop such traditions later on, and a lot of other philosophical schools besides... but that was after it had conquered other powers already. It used the existing institutional frameworks of the civilisations it had overrun with brute force to refine itself. It did this very successfully. But without the intial conquests, this would not have been possible.
Bottom line: unless Islam dramatically evolves into a more Buddhist-like form of religion, I suspect it will remain "just" a local Arabian tradition, without much chances to effectively spread outward. ...Unless it should be into East Africa, perhaps...!