You've misread, or the author misstated. Newton's law of gravitation, after accounting for the non-spherical nature of the planets, is incredibly accurate. NASA and other space agencies still use Newtonian gravity to plan, monitor, and control their space missions.
What the lunar laser ranging experiments did do was to test one of the key tenets of general relativity, the equivalence principle. The Moon has a rather weird geometry. The far side of the Moon has a much thicker crust than the near side. This results in about a two kilometer offset between the Moon's center of mass and its geometric center. It is this offset that gives the laser ranging experiments the ability to test the equivalence principle to an amazing degree of precision.
The best place in the solar system to look to to see the difference between Newtonian mechanics and general relativity is not the Moon. It is the orbit of Mercury. Suppose Mercury was the only planet in the solar system. Newtonian mechanics would indicate that Mercury would return to exactly the same spot in space after one orbit. General relativity says otherwise. It says that Mercury's orbit would like something like this:
The orbit is still more or less elliptical, but the perihelion (the point of the closest approach to the Sun) advances with each orbit. The portrayal in the above image is greatly exaggerated. Imagine a slightly larger scale model of Mercury's orbit such that the perihelion distance is one meter. Instead of four orbits as depicted above, imagine the precession over 415 or so orbits (100 years). Imagine placing a meter stick that goes from the center of the Sun to Mercury's perihelion at the start of this 100 year period, another meter stick that goes from the center of the Sun to Mercury's perihelion at the end of this 100 year period. Per general relativity, the angle subtended by those two meter sticks will be about 43 arcseconds (and this agrees with observations). That 43 seconds of arc: That is the thickness of a postcard between the tips those two meter sticks. Tiny!
Bottom line: Even for the orbit of Mercury, Newtonian mechanics misses the mark by a tiny, tiny bit over the course of a century. One has to look at the solar system for a very long time, or look outside the solar system at huge masses orbiting very closely to one another, before you can say that Newtonian mechanics "is not so accurate".
Those who claim that general relativity falsifies Newtonian mechanics have missed the mark. It doesn't. What general relativity did do was to show that Newtonian mechanics is not universally true. Newtonian mechanics is instead merely an approximation of reality. Then again, this is most likely the case for general relativity as well. Physicists have yet to reconcile general relativity with quantum mechanics, and when they do, the combined theory will have to differ from both in some subtle regard.
To me, the most profound difference between Newtonian mechanics and relativity is how the two look at space and time. Newtonian mechanics postulates that space and time are rigid, absolute, universal, and disconnected from one another. Relativity theory says otherwise. It is this "otherwise" that forces us to look at the universe in a very different regard. The accuracy (or lack thereof) of Newtonian mechanics with regard to lunar laser ranging experiments, or even the orbit of Mercury, is very much secondary to this huge mental shift in the nature of space nd time.