The answer is he didn't. He built on traditions, methods, and texts that preceded him, just as any modern scientist does. In fact, Newton himself famously wrote "If I have seen further it is by standing on the shoulders of giants." This is even more apparent when you consider that Gottfried Leibniz independently arrived at a (much more useful, it turned out) formulation of calculus.

Also keep in mind that the calculus of Newton is dramatically different than the calculus of today. Newton's calculus is more properly described as a collection of computational methods, rather than a unified theory. It wasn't until the 19th century that we see modern calculus, thanks to people like Cauchy and Weierstrass (who famously provided a formal definition of the limit) working to provide a solid theoretical foundation.

That's the historical side of things, if you're interested in the mathematical side I could go into that as well, but it might be outside the scope of this subreddit.