Can science prove or disprove the existence of God?

I'd avoid using the words "prove" and "disprove", because, outside the domain of mathematics, they tend to promote (or indicate) a misunderstanding of how rational evidence works. That said, I do think it is perfectly appropriate to say that God is scientifically unsupported (in the manner of the Teapot, the Flying Spaghetti Monster, the Invisible Pink Unicorn, the dragon in your garage, etc.), and that you must not have a fully rational worldview if you can permit yourself to believe in God.

If, under any given person's definition of God, they can think of some way the world would be empirically different if there were no God, and that actually does turn out to be the case, then that is scientific evidence against God. If they -can't- think of any way the world would be different if their God were not real — or if they refuse to think about or say what the differences would be — or if they claim that some observation would be evidence for God, but refuse to allow that its absence would be evidence against God — then they are delusional.

It is odd to say that science can investigate the existence of an individual thing, as odd as it would be to say that science could search for the best spouse for an individual person - but that is because it is usually not bothered with in such cases. Science - the principle that ideas should be tested by experiment - can be applied in any situation, and has been convincingly more powerful than almost all other methods of epistemology ever since its invention. The existence of anything which is in any way causally connected with us has implications about what will occur to us, and therefore can be tested by creating situations where the output would be different if the causal connection were absent.

Can science prove or disprove the existence of the Flying Spaghetti Monster?

There are so many "gods" with so many man made attributes that are entirely contradictory that the overwhelming evidence is against the existence of a single unifying god.

Gravity may only be a theory, but it gets us to the moon and other planets.

Fantastic claims require fantastic proofs.

Logic can be used to disprove certain depictions of God, however the scientific method is by definition agnostic.

To be termed scientific, a method of inquiry must be based on gathering observable, empirical and measurable evidence subject to specific principles of reasoning. A scientific method consists of the collection of data through observation and experimentation, and the formulation and testing of hypotheses.

God is often seen as being, by definition, unobservable, unmeasurable, and above the rules of science as the very creator of those rules.

How can the scientific method operate when it has nothing to work with?

I can understand people saying that they cannot agree with the concept of God logically, but the scientific method has nothing to work with.

The questions which science is incapable of proving and disproving fall into two categories: isolated incidents (who stole the cookie from the cookie jar?) and unphysical counterfactuals (could Batman beat Superman?). Regarding the possible persistent presence of a causally active intelligent being, science should be as effective at judging the existence of God as the existence of Mahmoud Ahmadinejad (to give an example of a real person) or Peter Parker (to give an example of a fictional one).