Citation needed? That's ridiculous. The empirical evidence is well over century old at this point. Start with the anomalous precession of Mercury's perihelion. That already can't be accounted for by Newtonian gravity.
I don't think they're saying the relativistic effects don't exist, just that they're still largely unimportant compared to Newtonian effects.
For precession of perihelion of Mercury we mostly noticed because any error is cumulative over time and we could integrate over an arbitrarily wide timebase. The relativistic effects are <10^-8 of the total, around 1/10th of the change imparted by Newtonian gravity of planets much, much further away. The BepiColombo orbiter should allow us to correct for the relativistic effects of other planets' pull on Mercury, but it's expected to be a change of <10^-12.
So I guess "many, many decimal places" is in the ballpark of 6-12.
Samsartor seems to think that the inverse square law does not hold at short distances (e.g. between the sun and mercury). Meindnoch agrees with mainstream physics that the inverse square law does indeed hold at short distances. You're confusing newtonian physics (busted) with the inverse square strength of gravity (still strongly supported); those are two different things. GR says gravity should be strictly 1/r^2, and this is what we observe in the solar system.
There are (ȓ/r^3) terms involving unit vectors, but that works out to 1/r^2 in practice. There are cubed terms in string theory and Quantum General Relativity (QGR) / Loop Quantum Gravity, but these do not apply at macroscopic distances. If you know of a url link to a non-theoretical inverse-cube effect which has actually been confirmed in lab experiments or actual observation, please post it.
