In particular, I'm told that people proved false theorems using Newton's approach to calculus.

I'm intrigued by this claim, but it seems very vague, both because there is no information about what the theorems might be and because it's not clear how widely accepted these false results are claimed to have been.

First off, I don't think it makes much sense to consider Newton in isolation from Leibniz. Newton didn't fully develop the calculus. His results were scattered in a variety of places and are not complete or systematic.

The rigor and logical validity of Newton and Leibniz's calculus were vigorously debated from very early on. Archbishop Berkeley published The Analyst in 1734, seven years after Newton's death. He claimed that although Newton's results were correct, they were derived through incorrect methods: "I have no Controversy about your Conclusions, but only about your Logic and Method." This would seem to suggest that if otherwise competent people arrived at incorrect and widely known results based on the Newton-Leibniz methods, it didn't happen during Newton's generation or the generation after that, since clearly a rabid critic like Berkeley would have made a big deal out of such a thing.

Blaszczyk at al. have an interesting revisionist take on the early history of the calculus. In their reading:

Leibniz’s heuristic law of continuity was implemented mathematically as Los’s theorem and later as the transfer principle over the hyperreals ..., while Leibniz’s heuristic law of homogeneity... was implemented mathematically as the standard part function ...

If you buy this account, then Newton-Leibniz calculus had a fully formed, well-defined, and consistent set of methods that are isomorphic to some subset of the methods of NSA. For example, say we calculate, in the Leibniz style,

$$d(x^2)/dx=[(x+dx)^2-x^2]/dx=(2x\,dx+dx^2)/dx=2x+dx{}_{\ulcorner\!\urcorner}2x.$$

The symbol ${}_{\ulcorner\!\urcorner}$ is Leibniz's notation for what he called "adequality," which Blaszczyk argues is the same as NSA's standard-part relation; today we'd write the final step as $\operatorname{st}(2x+dx)=2x$. This is a completely valid calculation if interpreted in terms of NSA. Of course the notation ${}_{\ulcorner\!\urcorner}$ never caught on, and practitioners of the calculus traditionally just wrote $=$, which made the practice of discarding squares of infinitesimals seem logically suspect. But that doesn't mean that the methods were wrong, just that they were traditionally written in a way that may have obscured their correctness.

Blaszczyk, Katz, and Sherry, "Ten Misconceptions from the History of Analysis and Their Debunking," 2012, http://arxiv.org/abs/1202.4153