According to current quantum theory, nothing would stop the inflation phase. Quantum variablility would cause it to continue randomly, with some areas not inflating, while others continue to do so.
Last time I read about this, the inflating areas would very quickly take over the entire Universe, and what you would have is, essentially, permanent inflation with ever-decreasing (in relative size) area of no inflation.
Obvioulsy, that's not happening. Also obviously, you can blame this problem on outr lack of full understanding of reality.
On the other hand, though, it certainly indicates that we absolutely don't know enough about it to make a statement like "The Big Bang didn't need God," which is merely technically correct because the statement doesn't mention inflation.
Without inflation, the Big Bang would have been no big whoop.
Lastly, it seems a little disingenuous for Hawking to poke fun at people by sarcastically asking "what was God doing before the Divine Creation," while at the same talk he went into M-Theory as a possible answer to how the Universe came to be.
M-Theory postulates time before the Big Bang - just like the religious minded do. The M-Theory time is not the time we experience, however, it is the time experienced in the multiverse where the membranes (branes) are floating around and occasionally knocking into each other (causing big bangs - but not inflation.)