I suppose it depends on what you mean by "science" Taken narrowly, no, science cannot prove the existence of God, insofar as we take science as being something distinguished from natural philosophy. For that matter, God is neither a subject nor an object of natural philosophy either. When God's existence is shown in natural philosophy it is as something outside natural philosophy, but which is a necessary cause of the things natural philosophy studies. Likewise, science can discover things which cannot be explained within its own discipline and therefore lead to a proof of God as a principle of the things science studies.
And once you speak of proving things about God, you are certainly in a different formality than either science (taken strictly) or natural philosophy (which may be called science taken broadly). At that point you are in the formality of metaphysics. Insofar as you make arguments a posteriori, it is true that the argument has a basis in say natural philosophy, but the argument itself is outside that formality.
This is something that, unfortunately, many modern thinkers are not attentive to. It is why an argument based on the Categories alone does not prove there cannot be absolute space (as that is in the formality of logic). We need natural philosophy and metaphysics to do that. It is the error that positivists make, or in general the whole modern project which wants to unify all disciplines under "science" meaning having mathematical certitude and models, etc.
So I think Dominic is right, if we take science, philosophy, etc in strict senses of the words, though it is legitimate to say (at least in principle) that things science holds can go as far as even demanding that God exists, though the argument is at that point in the realm of philosophy.
Though I would hasten to add, and I think pax would have to agree if I understand him right, that these arguments would be provisional, and not demonstrative proofs in the way, say the 5 ways present themselves as being, insofar as they are arguments from theories and hypothesises which themselves are subject to revision. An argument for God based, e.g. on relativity, depnds on that theory of relativity being true. Insofar as, as with most scientific theories, it is provisional, so is the argument from it
ETA: I am not dismissing provisional arguments. A good example is one Aquinas gives in the Summa contra gentiles. He shows that under the cosmology of those who held an eternal world there would have to be a God. This, even though he rejects an eternal world. Basically pointing out that even if they were right, God must be. A fortiori if the world is not eternal
εἰ ἐμὲ ἐδίωξαν, καὶ ὑμᾶς διώξουσιν: εἰ τὸν λόγον μου ἐτήρησαν, καὶ τὸν ὑμέτερον τηρήσουσιν.