It is hard to argue that Diamond does not prove his thesis. His meticulous documentation of evidence and his clear explanation of the meaning of that evidence make a very convincing case.
Diamond clearly lays out his claims and backs up every step with proof. He argues that early food production (along with diffusion and sharing) leads to the creation of complex societies. He argues that such societies provide the agents of conquest that allow them to dominate others. Diamond then shows that early food production was caused by the presence of suitable plants and animals. He uses scientific data to prove that the areas that had more such plants and animals got food production first.
Later in the book, Diamond also disproves arguments that are meant to refute his point. He shows, for example, how Europeans were not really more able than Aborigines to create a civilization in Australia. Thus, Diamond both proves his own points and refutes those arguments that disagree with his points. In this way, he clearly proves his thesis.
While Jared Diamond's thesis is somewhat controversial, as it pertains to the climate and its effect on food production (and in turn on human size, time for other activities, and societal organization), he does prove his thesis. In fact, some have criticized that he spent too much time proving one point by looking at several regions around the world, while ignoring other factors that contributed to why some societies developed differently than others.