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Is there even [a] way to [scientifically] pro[ve] God's existence?

Science is often thought to proceed by our logically deducing the laws that govern the world.  But it's not that simple; there are limits to what we can deduce, especially about things in which we cannot directly participate.  Radioactive decay is a good example of this.  We can't use a microscope to watch the events that make an element decay.  The process is quite mysterious.  But what we can do is make a simple theory of how decay might work, and then use that theory to make a prediction of what measurements we can expect.  That's the way science proceeds: by making theories that lead to predictions.  Sometimes these predictions turn out to be wrong.  That's fine: it means we must tinker with the theory, perhaps discard it outright, or maybe realise that it's completely okay under certain limited circumstances.  The hallmark of a good scientific theory is not what it seems to explain, but rather what it predicts.  After all, a theory that says the universe just appeared yesterday, complete with life on earth, fossils and so on, in a sense "explains" everything beautifully by simply defining it to be so; but it predicts absolutely nothing.  So from a scientific point of view it is not a very useful theory, because it contains nothing that allows its truth to be tested.  On the other hand, while it's arguable that the theory of quantum mechanics explains anything at all, it certainly does predict a huge number of different phenomena that have been observed; and that's what makes it a very useful theory.

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So certainly physics has not proven, and can never prove, that its theory of atomic decay is true.  The logical process is that if atoms decay randomly, then Poisson statistics will result.  Experiments show that Poisson statistics do indeed result, but logically this does not mean that atoms decay randomly.  Nevertheless, the way of science is that we do postulate that atoms decay randomly, until a new experiment calls this into question.  But no experiment ever has.  If this sounds like a reverse use of logic, then consider the same ideas for mechanics.  Ideas of gravity, mass and acceleration were originally produced by Newton through the same process: because they predicted planetary orbital periods that could be verified experimentally.  Because of this great success, expressions such as F = ma and F = GMm/r2 came to be canonical in physics.  The logic was indeed being used in reverse; but no one was surprised when, three centuries later, one of the moon astronauts dropped a feather and a hammer together in the moon's vacuum, and found that they both fell at the same rate (although it was still beautiful and dramatic to watch!).  That reverse logic had, after all, allowed him to get to the moon in the first place.  So this way of conducting science works very well.