Shakespeare's religious orientation and beliefs are an endless source of speculation and bickering among those who take an interest in the Bard. And for good reason: because Shakespeare's plays are replete with religious imagery, language, and themes, discerning a coherent and consistent worldview is exceptionally complex.
Robert Miola, a professor of English Literature, has written one of the best short introductions to the issue I have read in the recent issue of First Things. The traditional line, Miola contends, on Shakespeare's religiosity is that it is shaped by the Protestant Reformation and Protestant theology. That is, Shakespeare stands the quintessential Englishman, who has the tempered and moderate Protestantism of Elizabeth's via media.
However, in recent years that thesis has been challenged by those who contend that Shakespeare is smuggling Catholicism into England through his play. Miola rejects the "closet Catholic" hypothesis on grounds that it is a little too similar to a conspiracy theory. However, Miola makes a strong case for "evocative patterns of literary Catholicity in the bard’s work."
Yet while Miola is convincing, pitting Catholicism against Protestantism in this way may not be the most productive way of reading Shakespeare. It is plausible that Shakespeare was attempting to preserve a rational appreciation for the goodness of Roman Catholicism, while still acknowledging the abuses that had occurred. Shakespeare's world is very much like the King James Bible, which, as Adam Nicolson puts it, "does not choose. It absorbs and includes. It is in that sense catholic, as Jacobean Englishmen consistently called their church: not Roman but catholic, embracing all.”
One more point: Miola (rightly) demurs on the question of Shakespeare's private beliefs. The leap from text to author is a dangerous leap indeed. However, it is not a stretch to say that Shakespeare's plays are fundamentally Christian, regardless of their theological nuances. Indeed, Miola acknowledges an attempt to interpret Shakespeare as an atheist, but dismisses it as an "eccentric variation." For Miola, it seems, the weight of scholarly literature is firmly on the side of some sort of Christian underpinning to Shakespeare's dramatic world.