I think it can be proven historically that he was raised Catholic-- certainly the area in which he was raised was predominantly Catholic. However, England during the time when he was writing still had a cloud of anti-Catholicism (particularly in the aftermath of Bloody Mary and her raging lunacy), and although Elizabeth had brought religious conflict to a semblance of peace, you still couldn't just go out and be Catholic, straight up.

Since we know he was raised that way, I think it's safe to say it heavily influences his work, particularly Hamlet, Richard II, and The Tempest. And since his Catholic background influences him all the way up to his final play, I think it's safe to say he kept the ideals of Catholicism close to his heart, even if he wasn't a truly devout Catholic and he incorporated elements in his plays that might dwell outside of the realm of Catholic belief.

However, in terms of King Lear, I'd say that it's less Calvinism and more heavily influenced by the passing of Elizabeth. Shakespeare was almost certainly acquainted with Elizabeth, and I would say felt a certain fondness for her-- she was after all his most famous fan, and she was the queen during Shakespeare's entire life up to that point. There is a notable shift in Shakespeare's stylistic choices, and it's around this time that he wrote the great tragedies. He shows a certain contempt for royalty for a period while Elizabeth was dying and after her death when James took the throne. I'd say especially in King Lear, this helplessness that can certainly be interpreted as a type of Calvinism is a result of the changing of the throne and the general uncertainty in the air if England could be run as smoothly as it did under Elizabeth (note that three of the four great tragedies deal heavily with succession of kingship and the problems that come with it).

Long story short...yes, I think he was Catholic. LOL. But I doubt he was a strict Catholic, or even a Catholic who intentionally tried to put his religious beliefs into his plays, since he probably would have known better regarding the country's feelings towards Catholics at the time.