With Newtonian mechanics, and the appropriate mapping assumptions (how ellipses, foci, and differential equations relate to planets and stars) you can derive predictions about the orbits of the planets. The problem was, it didn't quite work. Newtonian physics failed to account for the orbit of first Saturn, and later Uranus. Astronomers at first assumed their orbital measurements were inaccurate (bad mapping assumption). Later, as orbital measurements improved an auxiliary hypothesis in the form of a conjectured new planet was introduced. This auxiliary hypothesis was tested, and found to be correct in what stands as one of the most spectacular predictions ever made by a scientific theory.
Simply put, Newtonian mechanics it failed in an early prediction. It had clear unambiguous observational consequences, and they were not observed. By naïve falsification it should have been abandoned. Obviously, it wasn't, but not because passed all tests or solved all outstanding problems. In fact, as Kuhn and others have pointed out, when it came to the formation of the solar system, Newton initially predicted less than competing theories. Newtonian mechanics succeeded, however, because it was very good at solving many problems, and it proposed a uniform way in which many more problems could be solved. In short, it was a good, general theory with wide potential application.
