Disproving God is impossible by all scientific methods. Many scientific theories are based upon evidence, but equally important is the lack of contradicting evidence. A theory is sometimes only considered "proven" by the common-man's perspective when it has not been disproven over a significant amount of time.
This is the kind of thing religious people talk about a lot, but I feel like this is a huge misunderstanding about the way science works, and the way people use words like "prove" or "disprove", and especially "theory". A theory is not a random guess, or a hypothesis. In science, a theory is a very solid model, highly agreed upon by near-unanimous consensus, supported by all available evidence. All scientific theories are rigorously tested by the scientific community before they can ascend to the status of "theory". Theory is the top of the food chain in science for large models. You don't "prove" a theory; a theory can't graduate and become something "more true" or better proven than a theory. It's unfortunate that scientists use the word "theory", because it also has a second meaning that many people confuse it with: a guess. A scientific theory is not the same thing as a guess.
So a scientist could not say there are different astronomical constants and have that be an actual Theory. A theory needs to have a large body of supporting evidence, peer review, and agreement by consensus that all evidence available points to the theory holding true against all applicable domains and tests. There is absolutely nothing that points towards relativity being weird at different spots in the universe; in fact, there is a huge amount of data that says that it's impossible for it to work like that. Space-time is a fabric, a unified thing, if it were actually a bunch of separate space-times (which is what it would have to be if different laws of physics were in different sections), then the entire universe would look much different.
Some of this argument is basically "Russel's Teapot" (see wikipedia). He said that he could claim that there was a teapot floating around a planet ten billion light years away, and that since no one can see that far, no one could disprove him. Some of this confusion comes with the word "disprove". First, let me go back to science: in science, it's not enough to make something up and have it hard to disprove. Science does not work that way at all. See The Scientific Method or wiki "science" in general to understand how ideas are advanced in science. Usually the word "disprove" is used (in science, and in many cases) to mean the act of showing that something previously thought proved is actually false. Scientifically speaking, you can't "disprove" something that has never been assumed proven. You don't need to - science does not deal with random ideas that have no basis in reality.
There's a problem too where some people think "disprove" means "convince me that I'm wrong", and that's entirely a social problem. I could say that the sky is green, and you could sit there yelling at me saying that it's blue all you want, but I could still say you haven't "disproved" that it's green just because I can be as stubborn as I want to be. Scientifically, nearly every definition of god has been "disproved", but people keep changing the definition or covering their ears and saying that it's not true. There's really not much that can be done about that. If we take "disprove" to mean "convince a hardcore christian", then it has nothing to do with logic, science, or physics. It's only psychology.
Just be a good person.
This, at least, I can agree with. However, I still think religion is part of the problem. As long as people think that there is some ultimate moral authority telling them what to do, who conveniently never actually says anything and has to have his commands relayed by politicians and priests, people will use this as a crutch to not have to think through their own actions. Actions should be thought through carefully, with consideration towards what effect it will have on other people. People should determine the moral worth of an action based on compassion, reason, and careful analysis. Too many people (not all, of course) simply take their orders from religious authority and walk away with a clean conscience without ever putting any thought into the actual effects their actions have on others around them. That's the main problem I have with religion - the entire reason it exists is as a shortcut for dealing with moral quandries and guilt.
Many of the main battles on the religious front in the U.S. today are in fact about stopping careful analysis or learning about the state of the world. Christianity is trying to shut down biology in schools because evolution isn't convenient to them. They are trying to shut down sexual education because it makes them uncomfortable. They are trying to shut down access to free media because they worry about what people will do if they hear non-Christian points of view. They are constantly trying to ban, censor, and discredit information that might hurt their faith. This is not the way to lead a moral life - the only way anyone can make a truly moral decision is to try to be as well-informed as they can about what is going on, and then decide what course of action to take. Purposely blinding yourself to outside information and refusing to learn is, to me, as immoral as willingly choosing to hurt others.