Supernatural elements are certainly recurring themes across Shakespeare. However, this alone cannot establish whether Shakespeare himself believed or disbelieved in the supernatural: only that he made use of it in his various works.
That being said, I would suggest that you keep in mind that Shakespeare lived from 1564 to 1616. This was the era of the Wars of Religion and where witch-hunts were a regular occurrence. Additionally, Shakespeare was contemporaneous with the likes of Galileo and Sir Francis Bacon—significantly before Newton, whose Principia wouldn't be published until 1687.
Shakespeare therefore lived in a time where modern science was still very much establishing itself (as opposed to the modern age, where it has become the dominant paradigm through which natural phenomena is viewed).
The Early Modern Era, in many respects, seems like a very alien place. People commonly believed in supernatural forces, and this was tied up with intense religious devotionalism, which shaped how people tended to think about the world around them. It is very likely, then, that this mindset influenced Shakespeare and his works.