So far as society is concerned, I don't think it really matters whether it's scientifically proven or not. What matters is whether people believe it and act accordingly. "Scientifically proven" means, more or less, that "scientists" believe it and act accordingly because of some experimental evidence that supports a particular hypothesis by refuting contradicting hypotheses.
So, either your hypothetical scientific proof is sufficient to convince everyone else to take up the proven religion, or it is not.
If the hypothetical scientific proof is not sufficient to convince non-scientists (at least, those not of the religion whose God is proved to exist), then you might have a situation a bit like popular belief in quantum mechanics or that smoking causes cancer. People on the whole believe it, kind of, because they mostly believe scientists tell the truth most of the time. But any given person might not understand it, might not act on their belief, or might refuse to believe it if that's inconvenient for them even though it's "scientifically proven". It could affect public policy (to the extent science ever does, i.e. to the extent it's convenient for politicians to believe scientists) without affecting every single person.
If it is enough to convince the layman, then everyone would believe in God. There might not be many historical societies we can look at where that was strictly true, but we certainly know of societies in which nearly all people broadly-speaking belong to the same religion, and where multiple religions exist but atheism is very rare. So the question is solved in the abstract: you'd have a society in which everyone sincerely believes in God, and that would be open to all the variety consistent with that.
Assuming no earth-shaking changes other than the common belief God exists, religions contain a lot of statements about the nature of God. It can be difficult in some religions even to get two randomly-selected co-religionists to agree a statement of belief, let alone between religions. So, depending on the nature of the proof there might be a lot for religions to disagree on, principally the practical consequences of the existence of God for morality, worship, and so on. So perhaps people could on the whole retain their religions, present-day atheists could reason that although it turns out God exists, that has no consequence for them, and so on.
So, just because God exists doesn't necessarily mean he rewards virtue, punishes evil, or that there's an afterlife, or that he wants people to be kind to animals, or anything else someone might advocate in the name of God. People of different religions can easily believe that God exists and carry on disagreeing about everything else. See for example the Protestant Reformation: the dispute was not over the existence of an omnipotent, omniscient God. Perhaps not quite everyone was agreed there was one, but I don't see things going down substantially differently if it had been literally everyone.
However, if the nature of the proof is, for example, "Christianity is proved correct because Judgement Day occurs", then of course Revelation has a lot to say about what happens around that event, which we could say is substantially correct in your world. Then we can say that society would be radically different from any that's gone before, on account of Heaven being manifest and whatnot.
If the world doesn't end but one religion is proved correct in general, not just specifically in its statement that God exists, then what would follow would be a society in which everyone sincerely believes that particular religion. The nature of that society depends very much on the nature and strictures of the religion you choose. And have we proved "the religion as a whole" correct, or have we proved that a particular sect or denomination is correct? The Roman Catholic society is different from the Methodist one, do we have to choose one or do we keep both? Basically, to prove "the Christian God" exists, you have to decide which of the properties ascribed to God by various Christians at various times, are actually possessed by (and proved of) the God who has been proved to exist.
The actual transition might be very complex and unruly. It's not necessary that everyone would just convert to (for example) Islam overnight. AFAIK it's possible to convert overnight, but people would have to take their time beforehand to process the proof and be convinced by it, and afterwards to learn how to live as a Muslim. Regardless of the practicalities of transition, if the case is presented convincingly that everyone should be a Muslim then the trend would be towards a Muslim society.
Other religions take different views on conversion. For example, I think (someone please correct me if I'm wrong) that I could believe Judaism to be true without necessarily concluding that I should become a Jew. Naturally you'd have to carefully research the specific religion that you decide to make true in your fiction, or you'll have an offensive pastiche at best ;-)
Finally, I note that there's a great deal of speculative fiction in which the existence of some kind of supernatural goings-on could be scientifically proven, except that it's rare and there's a conspiracy to keep it secret, and ironic or farcical plot elements combine to prevent the proof being documented, and so on. If the scientific proof of God was of this kind: "there are actual angels running around, and they've met God, so there's your proof" -- "well, I haven't met any angels yet, so I don't believe it yet", then you have plenty of fictional precedent for what might happen. Hi-jinks, basically.