A miracle is supposed to be distinguishable from random events. The idea is that a miracle is something extremely unlikely statistically and that it is the product of a divine decision.
Well in that case the Vatican could just hire scientists of any confession to authenticate miracles and use a peer-review process. A limb is growing back, a resurrection occurred, a true prophecy was revealed. There would be a list of scientifically confirmed miracles and of course scientists would try to understand what causes them and would finally have something to study in religion.
Demonic possession exists you say? Then it means that there is a kind of sentient entity able to control human bodies. If it is confirmed sentient, I would assume deception (and most religion say to assume that anyway with demons) so it would be a proof of "something": aliens, extra-dimensional beings, maybe real demons out of the christian bible, maybe beings posing as them. How does it work accross faiths? do they present themselves as djinns when possessing muslims? do they reveal testable information? Can they possess atheists? What are the invariants and the unknowns? That's a scientific process, yes, and that can be done with demonic possession.
I personally would admit the occurrence of these events and would definitely put back on the table hypothesis that I currently find unlikely: that the bible was a work of revelation, or the Koran, Torah, etc... It would now become crucially important to understand what these old revelations were and what they meant. We would definitely have to reset the whole exegesis and replace it with scientific and historically minded studies. Because, face it, even if the original revelations were divinely inspired, theology so far has been another word for making up bullshit. Getting the revelations correctly would become a history subject, not a theological one.
I guess this hypothesis is basically what happens in Evangelion: Divine revelations, angels and divine magic is real. It means we can dive in it and understand more about our destiny, maybe control it.
The thing is, I have a hard time finding a scenario where "God" would encompass anything sensible and observable. For any entity we could test and observe, we would have better words to describe it. Is it really a sentient being with an agency? Is it a law of physics? It is a bounded by anything (time, causality, space, morality)? Is it trustable? Is it acting randomly or with a plan? Do we share the same definition of morality if it has one? Does it asks for worship? Does it give reason for it? Is it the sole provider of the afterlife service? etc...
I feel that to describe any existing entity, the word God is as useless in 2015 as phlogiston is to describe chemical processes. Sure you can shoehorn it somewhere, but it does not really fit and you have tons of better alternatives.