I'm one of those people who think that you cannot prove the existence of God. However, what I mean by 'proof' is what is meant in a formal system where you can deductively demonstrate that a conjecture has been proven by definition to be true.
Proof in terms of being fully convinced, is another matter. This is like an inductive 'proof' that is not really a proof but merely highly convincing. For example, Einstein proved that E=mc^2, but this is not a formal proof, merely a highly reliable result of physics (an inductive process of finding out answers).
This second kind of proof might be possible for the existence of God. That is, it might someday be necessary to believe in God to best accommodate scientific results. If that were to occur, then the definition of God would probably be rather explicit and not favor most religious conceptions of God. One might even argue that the scientific acceptance is not even equivalent to the God of religion.
As far as Gödel's second incompleteness proof, this is strictly a formal system proof concerning certain branches of mathematics (e.g., set theory) that can only be interpreted as having something to do with the real world. The interpretation is always going to be done inductively (i.e., non-formally), and therefore the formal usage of Gödel's proof as it relates to the world is ineffective. In short, Gödel's proof can say nothing about the world in any formal/deductive sense. Any inductive interpretation has severe frailities (e.g., Hume's problem of induction, choice of inductive method, underdetermination of theories, etc). Therefore, at best the interpretation of Gödel's theorem to the world has limited application as it relates to applied mathematics.
Warm regards, Harv