The traditional account in Muslim sources from a couple centuries later is essentially "all of it" but that following his death a number of tribes committed apostasy, leaving the Muslim polity in a revolt that was quashed by Abu Bakr, known collectively as the Ridda Wars.

There are a number of reasons to believe that this may not have been the case, however, and that various parts of Arabia, particularly the eastern coast, may not have been under Muslim control until the 640's. Robert Hoyland's "In God's Path" is quite good on this.

The basic fact is that we have serious shortcomings in terms of sources about these kinds of early questions.