Inspired from the commentary on this question.
How far apart do two objects have to be for the gravitational force between them to be negligible? By negligible I mean, that it could never be measured rather than a technical limitation that results in a measurement/force of zero and/or the accuracy of the measurement is too imprecise to be of use.
Basically when is the force to small to ever be measured, now or in the future? Is there a limiting distance?
Answer
In classical gravity, the answer is "never". In general relativity, the answer is "never".
Now what about a quantum theory of gravity? We don't know how it'll work, but it should reduce to general relativity in the classical limit (i.e. the limit of weak fields and large distances, which is exactly what your question is about). So the answer is still "never".
What about when the force gets less then the "Planck force" or whatever? "The Planck foo" is completely meaningless; it's just the foo that can be formed out of $\hbar$, $c$, and $G$. The significance is that because it has all of those constants in it, the Planck foo might be a quantity that comes up in quantum gravity.
It turns out the Planck force is over $10^{44}$ Newtons, so it might be an indication that something weird happens when forces get that high. Again, it says nothing about what happens when forces get low; that should just continue to work.
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