Pete,

Its quite easy to verify gravity (the inverse square law, which goes back to Newton and Hooke of course) on familiar length scales, and at familiar accelerations. Such was done by Henry Cavendish in the early 19th century, and he (probably) measured the universal constant of gravitation, G. Also Newton's model explains very well motions on larger scales, such as the motions of the planets. Or nearly. It was found in the 19th century that the precession of the perihelion of the orbit of Mercury. Einstein's theory of General Relativity modifies Newtonian gravity at high accelerations (thats a bit of an oversimplification) and it models the precession of the perihelion of Mercury very well. And predicts some other things later verified by experiment such as the gravitational deflection of light (those photos of gravitational lenses that the Hubble Space Telescope produces).

So thats all very well, but what about the limit of low accelerations? Low accelerations mean large distances, so here you can't make experimental verification on earth, or even on the Solar System (though there is the Pioneer anomaly). But if you look at the motions of galaxies in clusters of galaxies, then Newton's laws clearly don't explain these, unless you postulate a large amount of hidden mass. Dark matter, which we have never detected directly, despite a variety of very expensive experiments to do so. So either there is dark matter or there is a modification to Newton's laws in the low acceleration limit, hence there is a variety of opinions.