That would be awesome! The catholic church certainly has the resources to do that kind of thing, and churches are already like a sort of social network, so they have a very captive audience. They are in a unique position to create a social network that doesn't need to monetize through any kind of ads or data collection, and I imagine their status as a religious organization can even get them tax breaks.
Reminds me of the book Anathem (great book, tons of Catholic imagery)