Stephen Hawking cannot say whether God exists. All he can say is that there is no need for us to assume that He does.
Try this scenario on for size:
The existence of the universe is accounted for by the postulates of the inflationary Big Bang. Alan Guth mathematically defines an “inflaton” particle that was a child of a highly excited gravitational field. This extremely excited and energetic inflaton particle, being much like an excited radioactive nucleus in an atom or an excited particle in a particle accelerator machine, had to decay eventually. When it did so, matter and energy, in types that we know, were formed. The inflaton particle was a point-particle, that is, a point-mass having mass, gravitational potential energy and all other types of energy equivalent to the matter and energy that we can tally today all rolled up inside.
For a black-hole F = GMm/kr, k = constant = 1m (SI System), for dimensional purity. The inflaton particle possessed a hyperbolic 1/kr gravitational field with gravity existing prior to the big bang. The prior existence is said because we live in a multiverse (See followers of John Archibald Wheeler)) and the inflaton was only one of many such particles that existed and do still exist in the meta-universe (see Hugh Everett).
The big bang happened. And, it will happen again for other inflaton particles and has already happened in many others of them, long “before” our inflaton decayed into our universe. In other words, we must postulate “meta-time” as well.
Since the gravitational field existed prior to decay of the inflaton, gravity being “leaky” (George Dvali, Lisa Randall), overflowing boundaries like those associated with an infinitely dense, infinitely intense gravitational field stemming from a point-mass, it had such a hyperbolic 1/kr potential profile. When the inflaton decayed, the 1/kr field began to collapse. It is still collapsing, transitioning to the 1/r^2 gravitational field that we see today. On a graph, we see that the equivalent hyperbolic potential energy curve is inherently greater than the equivalent 1/r^2 parabolic curve. So, the continuing collapse of the hyperbolic 1/kr field releases energy into the visible universe. The only way this energy can be manifested is by means of the expansion of spacetime.
This expansion is kinematically equivalent to our having all the discrete objects in the universe rushing away from each other. Most of the time, we should think about expansion kinematically to avoid confusion.
Now, a graphical look at hyperbolic exponential decay reveals that acceleration of expansion may still be occurring slowly, but it should be happening at an infinitesimally lower and lower rate (after all, expansion has been progressing for 13.72 billion years already and the manspring should have wound down quite a lot by now). This potential energy, bound up in the remaining remnant of the hyperbolic gravitational field, is real energy (with mass equivalent via E = mc^2). It accounts for the “missing mass” of the universe and the accelerating rate of expansion without invoking any new forces like quintessence.
The unique role of gravity in this scenario is a key. You might say that gravity is the only true force, all other forces being derived from it, and its status is fixed by this role. It is such a weak force and cannot easily fit into quantum theories because of its status as the “Ur” force. (Ur was the first true city in Western civilization).
This scenario is hypothetical, to be sure. But, it is no more speculative than Alan Guth’s original formulation of “Inflation” as a solution to the various paradoxes of the Big Bang theory. There is “speculation” and there is speculation. Some intriguing speculation is more robust and worthy than other kinds.
At any rate, we see here that there is no need for God in accounts of the origin of the universe. This does not mean that there is no God. But, if He exists, He must be infinitely bigger and more powerful than anyone ever thought. He presides over a multiverse, not a mere universe, and He has made meta-time as well as standard time. He must be truly almighty.