Well yes, we can. Though I suppose it depends on what you mean by "proof". It's very difficult, if not impossible, to ever stop someone from jumping up and rejecting any point you make, no matter how convincing it is. But there are a number of reasons that "prove" to beyond any reasonable doubt that the moon is not made of green cheese:
1. Its density would be different from what it is. cheese has an average density not much more than water. Since we can measure the mass of the moon as well as its size, we can say confidently that the moon is too dense to be made of cheese. Initially, at least, a cheese moon would have to have a radius of approximately 2,500km as opposed to 1,740km.
2. even though the outer surface of the moon is a vacuum, inside cheese-eating bacteria could, and would, happily live and eat away turning the cheese into something else. After such a long time all, or virtually all, of the cheese would have been consumed and turned into waste products. So it would have rotted from the inside out.
3. Someone's gone and calculated the effects of pressure on such a large mass of cheese -- find here:
http://www.stwing.upenn.edu/~fitz/humor/moon_cheese.html . If you can't be bothered to read the link, the point is that the cheese couldn't stay cheese for long and would turn into water + gases.
4. Anyway the moon has been hit by many objects over the eons (see all the craters). A cheese moon would have been blasted away, indeed possibly destroyed entirely, if it were still cheese - the temperature involved in a collision would have melted the cheese or even made it evaporate.
5. We've visited the surface and found that the surface, or at least that part which we landed on, is rock. As described above a cheese moon would turn into water + gases which is not what we see.
Anyway, a green cheese theory is completely unsustainable and fails to match the evidence in any way. Therefore it's "proven" experimentally that the Moon is not made of green cheese.
A silly post, perhaps, but there is a serious point here. Atheism, or any world view, shouldn't be about "can't be bothered wasting time trying to prove that we're right". It should be almost precisely the opposite. Looking at the evidence and forming a view of the World that is based on that evidence. If that evidence points convincingly and overwhelmingly towards the existence of a God, then you should acept that God's existence. If, on the other hand, the evidence cannot be fit into a theory including a God, then you should reject the existence of God.
Perhaps a case cannot be built either way, and it may stay that way forever. But even then, you should continue to try. It's not a matter of "oh, that's silly and I can't be bothered thinking about it". Or, at least, it shouldn't be. It shoudl be more rational than that, and accepting that the "Ultimate Truth" is how the World is, not how we think it should be.