While Newton’s gravitational laws like F= GM1M2/R^2 are used in Relativistic operations when they are accurate enough, due to Perturbation theory there are limits and exceptions when only G is really carried over when you need more accuracy than the classical model provides.
Remember that while a useful concept, gravity is a fictitious force or an apparent force that is due to being in an accelerated reference frame. There is also a limit you hit because gravity under Newtonian Physics acts instantaneously across the entire universe while we now know that it is limited by the speed of light or the speed of causality. When gravity is not used to simplify the math, it is the curvature of space that is the best accepted current model.
The General Relativity model works all the way up to black holes and neutron stars, and to an accuracy that if someone asked the distance from you to the moon, the error would be less then the distance from the top of your head to your eyebrows. Where it does fall down is when the numbers get very very very small, like trying to map the effects to the scale past our local group. But to be honest that probably has more to do with computers inaccuracies with really really small numbers than anything fundamental.
As others have mentioned, the most accurate tests will not be using the Newtonian model, but the largest test for the equivalence principle is probably this one fromJuly 4th in Nature.
While QM and GR are in conflict related to the quantization of gravity, and neither theory is *complete *due to this disagreement, these disagreeing theories are sufficient enough to explain why gold is the color it is, and why your wedding ring doesn’t explode when you wash your hands. This is directly related to Gold having more S-orbitals, and relativistic contraction impacting the energy levels for absorption which for most metals is in the UV range but is shifted to our visible range due to the curvature of spacetime (but it can answered in SR with less math). The tools work in useful ways and both GR and QFT are the most accurate models we have ever developed, and that link above pretty much destroyed the MOND theories by putting experimental tests results into way too narrow of a band for them to work. (that had already been done btw)
We do need a unifying theory, but the changes it will make to the changes to the answers of the math we already use very small.