First, of all it is necessary to establish something important. This is sometimes forgotten about in discussions like these. Skepticism is a method, it is not a belief system. Atheism is the default position from skepticism. If one is a skeptic, as everyone should be, it is only natural for the person to be an atheist. Therefore, the atheist does not need to demonstrate his position. The atheist has the default position based on lack of evidence or lack of argument. So Hawking cannot be challenged on the ground that he never demonstrates that God does not exist because he does not need to do such a demonstration.
One way in which science supports belief in God is that the laws of science themselves require a lawmaker.
This is the failed cosmological argument. The cosmological argument says that the universe is here therefore it had to come from somewhere because everything can from somewhere. It then concludes that this "somewhere" is God. But the conclusion of this argument contradicts its premise. The premise was that everything had to come from somewhere. So where did this mysterious God come from? It never answers its own self-contradiction. The theist simply asserts that God did not come from somewhere, God always existed. But why then can we not simply assume that Nature always existed? Perhaps Nature, its laws, and how they describe the world, always existed? It were these laws that formed the universe. The theist says that "God", the mysterious first cause, always existed. But they never say what God is. God is some mysterious entity never specified. The only property this "God" has is that it was the always existing main cause of everything. But if so then why not simply take the position that Nature has these properties. That way this "God" is simply Nature. And so there is nothing special to "God". Of course, equating God with Nature was what Spinoza did, if it makes you feel better you can do it, but for practical purposes this is atheism. Thus, the cosmological argument is a failed one. It fails because it leads to a self-contradiction and it fails because it never specifies what "God" means.
But the other problem with the cosmological argument, as every argument for God, is that it is only an argument for deism, not theism. I got no problem with deism. It is an understandable position for someone to take. If you simply wrote an article on deism then I would not have an issue with it. My problem is that people substitute theism for deism, which are far far apart. Deism simply asserts that the universe had an Architect for it, some external entity outside the universe that was intelligent in what it made. That is it! It does not say that this "God" (referring to this Architect) cares about people. It does not say that God has a relationship with other people. It does not say that God listens to other people. It does not say that God intervenes with the world. It does not say that God performs miracles. It does not say that God communicates with prophets. It does not say that there is a life after death. It does not say that the universe was created by God for us. For all purposes deism is basically atheism in every way. My problem with the cosmological arguments and teleological arguments is that people argue for deism, which is fine, but then they magically substitute the God of the Bible into the mysterious undefined "God". Which is not fine. There is a long path to go by to explain exactly how this "God" satisfies the conditions for the wicked God of the Bible. Believing in God and being religious are very different. You can believe in God but be extremely anti-religious, like Thomas Paine or Thomas Jefferson. So stop using this argument as if it leads to theism.
Einstein, no believer in a conscious God, nevertheless often expressed amazement at the comprehensibility of the universe.
And I too am very much amazed with the harmony and mathematical structure of the universe. I do believe there is a rational order to Nature. However, I do not assert this to be "God". For me it is just the way things are. They necessarily must be this way and no other, they necessarily must follow mathematical laws and therefore be ordered. Why must the order and harmony of the universe imply a God? Perhaps this is the way things are? Why must we invent hypothesis that are not necessary?
Even if Hawking is correct that the laws of gravity and quantum theory allow universes to appear spontaneously from nothing, that they somehow breathe fire into themselves, he has not explained how these laws themselves came to be legislated.
Hawking does not need to explain it. He can simply assert "I do not know", and this is okay. Atheists never pretend they know all the answers, in fact atheism is the humble position of doubt, not certainty that is found in all religions. Hawking might very well say, "I have no idea where the fundamental laws of nature came from". Just because we do not know the answer to this question does not imply a God. Using the excuse of not knowing to fall back on God is the old "God of the Gaps" argument. Every time there is a Gap in scientific knowledge the theist always falls back on that we do not know. First the theists asked where diseases come? When science answered that question they asked where life came from. When science answered that question they asked where the universe came from. When science answered that question they next asked where the fundamental laws came from. And this keeps on going and going for thousands of years. Religion never makes an effort to present an answer, it uses our ignorance as its strength. And this is exactly what God of the Gaps is. Every time we do not know we invent a God to explain it. And every time it has been done it was shown that such a God was unnecessary. So instead of saying that a God must have been behind the fundamental laws why it is not okay to simply assert that we do not know the answers and be done with it? Why not be humble for once?
The second way in which science is employed to give rational support for faith is that were the laws of nature to be different in the slightest way, our universe would not be possible. Some famous atheists such as Douglas Adams dismissed this argument, claiming that it is like a puddle marveling that its hole in the ground is exactly the right shape for it. But this entirely misses the fact that our universe is not any old universe, but rather an amazing universe that allows for the formation of such complex phenomena as matter, planetary systems, life, and intelligence.
Again an argument for deism not theism. But if you want a refutation for this argument just consider Christopher Hitchens' response to this argument. We are the only inhabited planet with intelligent life, actually just life, as far as we know. We are one of the few successful solar systems. Other solar systems have failed to developed. Other stars failed to become stars. Other galaxies fell apart. We are unique in that it went well for us, so far. But what about the millions or other parts of the universe which failed? What about those? It is standard human nature to look at what is seen and ignore what is unseen. We can easily focus on ourselves but what about all the unseen failures that happened in the universe? Why are we not focusing on those? It is only fair to compare failure to success and failure far exceeds success in our universe. This only shows that there is no plan for the universe. It may be ordered and rational, but it does not care what the final outcome is. We can be here one day, and dead the next, just like with the dinosaurs. It may be benevolent at one instance and malevolent the next. There is no grand plan for anything, and we are not here for any specific purpose.
In response to this, it is first important to note that the multiverse model is entirely speculative, with no actual evidence whatsoever.
But God is not speculative? There is actual evidence for God? What a double standard by Slifkin. When scientists propose an explanation which they cannot yet test they will acknowledge their inability to test it but present an argument to why they think it might be true. But the religious people will refute all their work by saying "you got no evidence for it". Though they will never ever do that to their precious God.