OP's question implies that currently religion CAN be proven. Which of course it can't. Finding ETI won't 'disprove' religion.
All mainstream religions were 'made' when humanity's knowledge of the true cosmos was nil, so the thinking applied to 'create' these religions belongs in a bygone era. Not many religions attempt to describe the physical universe on technical terms, and as such most scientific discoveries - that of Copernicus, Galileo, Hubble etc - are incorporated into religious thinking and everyone gets on with their life. Religion's concepts of creation are widely accepted, given the sleuth of discoveries that contradict them, as being symbolic and poetic; they are not and never claimed to be factual.
The issue for Christianity is thus: Christ took the form of a human to save us. A human. He didn't take the form of a cat to save the cats, or a parrot to save the parrots, he became a human as Christianity's main focus is on our race; it doesn't seem to be able to account for other intelligent races.
Christianity does have an escape clause, however, in claiming that there may be multiple saviors for multiple alien races. Precisely one 'Jesus' per deserving alien race. However the bible states that
our Jesus is God's
only son, and many devout Christians would call the idea of there being more than one heresy. However the discovery of ETI, still won't be a problem and certainly won't 'end' religion; in fact it probably won't even make a dent.
This survey shows that almost all people who identify themselves as being of one of the mainstream religions aren't concerned at all about the implications of finding ETI and the effect they would have on their belief. The biggest possible blow to religions that herald humanity as being unique was evolution. Evolution was a harder pill to swallow for theologists than finding ETI will be, as it took their view of humans being saintly and unique and threw it in a blender. Religion, most notably western Christianity, struggled - and still is struggling - to assimilate evolution into their theology. But they keep on. They take the awkward scientific discovery, look at what it disproves, and then say that part of their religious text is symbolic or metaphorical and isn't to be taken verbatim.
Most religions will keep adapting to the point that their texts are essentially just 'guides' to living a decent, morally sound life. However, I believe some will become very unnerved when repeated scientific discoveries seem to disprove part after part of their religion and it will become harder for younger generations to accept the theology of their ancestors and a lot of religions will disappear, probably within a few thousand years. However some will stick around, probably forever.