I think one of the problems is that we tend to read in a postmodern sense, meaning through our own pride and prejudice. In other words, we want to see Shakespeare reflecting back to us our own understanding of the world. So those people opposed to Catholicism haven’t grappled with or fathomed that dimension.
I can see where people might respond by saying I’m the pot calling the kettle black because I’m a Catholic myself, but I insist in my scholarship on reading the work objectively through the eyes of the author, which means we have to learn as much as possible about the author. So although you can prove Shakespeare’s Catholicism by reading the plays, I think you have to begin by proving it biographically and historically through his life, and I think there’s enough evidence now beyond any reasonable doubt that Shakespeare was certainly a Catholic in sympathy and to one degree or another a practicing, recusant Catholic.
Writer Joseph Pearce on the case for Shakespeare’s Catholicism
Shakespeare Was Catholic!