When does Newtonian physics fail?
The answer by Zo the Relativist to the question How accurate is Newtonian Gravity? includes the statement:
The key point is that Newtonian physics fails when, roughly, the quantity $v/c\ > 0.1$ or $GM/c^2r > 0.1$.
This was stated on this exchange 11 years ago. Where is the proof for this, if it's even correct?
"Fail" is a relative term. For some applications a 1% error is enough to say the theory is failing, for others getting roughly the right order of magnitude is a success.
Special relativity implies that momentum is $p = mv/\sqrt{1-v^2/c^2}$, so the difference between it and classical momentum $mv$ grows as $$mv\left(\frac{1}{\sqrt{1-v^2/c^2}}-1\right)\approx mv \left(\frac{1}{2}\left(\frac{v}{c}\right)^2 +\frac{3}{8}\left(\frac{v}{c}\right)^4 +\frac{5}{16}\left(\frac{v}{c}\right)^6\ldots \right).$$ So the relative error starts growing as $(v/c)^2/2$. If you demand better than 1% precision, you are safe when $v < 0.14c$.
The other failure direction is when general relativity starts to bite. The rule of thumb that when $GM/c^2R$ no longer is small implies that the system is approaching the size of its Schwarzschild radius. The precession of an orbiting body is $\Delta \phi \approx 6\pi G(M+m)/c^2 a (1-e^2)$: again there is a roughly $GM/c^2R$ dependency, and the difference between the Newtonian and relativistic answer starts to matter when that factor no longer can be approximated as zero - but in the famous example of Mercury and the sun, the distance is about 22 million Schwarzschild radii, so here it was more about the precision of measurement rather than behaving very differently from theory.
Another way of looking at it is the Newtonian limit of general relativity, where $g_{tt} = -1-2\phi(x)/c^2$, where $\phi(x)$ is the classical potential. This applies when $\phi(x)/c^2 \ll 1$. When that is no longer true we may suspect that other factors need to be included and the approximation will become bad.
But note that general relativity also predicts gravitational waves in the weak field limit, something Newtonian mechanics does not predict: while the theories approach each other for slow-moving not too dense masses, they do make different predictions about energy loss due to gravitational waves.
Rules of thumb are just rules of thumb. Think about your requirements before using them.
We have two completely different models of physics: Newton's, which is a very good approximation for everyday purposes, and Einstein's, which is an even better approximation and which is known to furnish experimentally verified right answers in circumstances where Newton's laws don't.
So, you take the predictions of both models for differing experimental conditions and compare them one-to-one. For everyday conditions (low energies, low velocities, small masses) the two yield identical answers within experimental error. But there comes a point where the two begin to significantly diverge: Einstein's model tracks the truth while Newton's starts going into the weeds.
This sets an approximate but handy and practical "usefulness limit" on Newton's model. There is no closed-form, precise proof of this; mathematically, one such limit is suggested in the quote you cite. But if you wish to specify some particular break point as, say, either 0.5%, 1%, 5% or 10% deviation between models, then you can solve for each case and declare the conditions present there at your case of choice to be the usefulness limit.
Regarding kinematics, and kinetic energy ($T$):
$$ T = E-mc^2 = \sqrt{((mv)c)^2 + (mc^2)^2} - mc^2 $$
So we with $\beta=v/c$ and doing the Taylor expansion around $\beta=0$:
$$T =\Big( \frac 1 2 \beta^2 - \frac 1 8 \beta^4 \Big)mc^2$$
$$ T = \frac 1 2 mv^2 \Big(1 - \frac 1 4 \beta^2\Big ) $$
so relativistic corrections start as:
$$ \Big(\frac v {2c}\Big)^2 $$
So at $v=0.1c$, that's a one-part-in-400 correction, but ofc, higher order terms enter rapidly as $v\rightarrow c$.
Gravity is a little more complicated. I posted it somewhere on this sight, from a paper, but the result is you can expand the Schwarzschild metric into a potential and factor out the newtonian part of each term and get something like:
$$ \Phi(r) = -\frac{GM}{r}\Big({\bf 1} + \alpha_2\big(\frac{r_S}r\big)^2 + \alpha_4\big(\frac{r_S}r\big)^4 + \cdots \Big) $$
where the $\alpha_i$ are "average" size, and the scale parameter, $r_S$, is the Schwarzschild radius.
When these are applied in the real world (aka: interplanetary spacecraft navigation), they're called parameterized post Newtonian formalism ( https://en.wikipedia.org/wiki/Parameterized_post-Newtonian_formalism) for massive bodies, and post-Minkowski expansion for high speeds (https://en.wikipedia.org/wiki/Post-Minkowskian_expansion) covering:
