God's existence is beyond any scientific fact, although logic could come to rescue.
As Thomas Aquinas, Blaise Pascal also addressed this issue.
When he chose to believe in God, it was just the most primitive self-preservation instinct manifested as an unconscious rational choice (pretty much a conscious one, in his case).
As Blaise Pascal stated (see Pascal's Wager): He chose to believe in God as there is no proof that God doesn't exist. So, should God exist, his reward would be heaven and eternal joy. Should God not exist, he didn't lose anything.
In our days, insurance works more or less the same way. You choose to have insurance because it's better to have one though you're possibly never going to use it than to need one and not to have it.
Eventually a hypothetical religion could emerge, with a divinity that could create "everything" in such a way that proof of that divinity's existence could never be found. In which case Pascal is 100% right... again. Thus, atheists could not blame believers of that religion, as it is their most rational response to the uncertainty (though believers could blame atheists trying to find out whether God exists, as being somewhat masochistic, from their point of view of course).
Regarding that hypothetical religion one thing is for certain: believers and atheist both could agree that, if such a divinity exist, it is more likely that they're never going to find any proof of its existence.
I'd like to call this hypothetical religion "Religion 2.0", where even science could be considered part of its rituals, and every attempt to prove the non-existence of their God could be celebrated and glorified.
As a premise the question named God, no whose God, so in the case of religion 2.0 (the example above), Hawkings' argument is not sufficient as God could create an eternal universe, in the same way as one with a begining.
