But... well, for one thing, if there were demons that were only affected by specifically Christian ritual, that would be a strong argument for the correctness of the Christian faith. So first of all, respect for being willing to stick to your guns even in the presence of hypothetical concrete evidence against it.

Second, demons and hauntings aren't a particularly Christian thing, they're just depicted in that way in those specific movies. What rituals would work would depend on what the demons or ghosts were actually like. There's every chance they would be Jewish demons, or demons not associated with any religion, or whatever.

Third - and I hesitate to say it - but demons and ghosts aren't real.