I think it would. Instantly. In fact, I hope one day soon that we somehow encounter extraterrestrial life because I think there would be a wholesale REJECTION of the bible and any/all religious institutions.
How do you figure that? There is nothing in the bible denying the existence of extra terrestrial life so why would such a discovery pose such a huge problem for religion? By contrast, Genesis does implicitly deny the evolution of different species and yet the discovery of evolution didn't result in a wholesale rejection of the bible and any/all religious institutions, did it?
If religious people can turn a blind eye to the far more troublesome, anti-biblical reality of evolution, what makes you think they will be phased by the discovery of ET life. They will just spin it by saying the ET life is part of God's creation that was made as part of the heavens and that simply isn't mentioned in the bible - just as a lot of other things aren't mentioned in the bible. They would say the bible doesn't mention ET life because it simply isn't relevant to the theme of the bible and that the bible is focusing on issues pertaining to earth and man's salvation.
As a matter of fact, the Seventh Day Adventists believe in the existence of other "unfallen" worlds - other planets inhabited by intelligent beings. It is not something they speak about much, but it is in their literature.