Would Newton law of gravity be the same between two neutrons at smaller distance? Imagine 2 neutrons placed at some distance apart, they should only gravitationally attract each other according to Newton law of gravity. Theoretically the law should work at smaller distance until Pauli exclusion principle set in, how about experimentally?
 A: Neutrons have a magnetic moment on the order of $\mu=e\hbar/m$ and therefore experience an electromagnetic force of order $\mu_0 \mu^2/r^4$ or $\mu_0 e^2 \hbar^2/m^2 r^4$. Their gravitational force is of course $Gm^2/r^2$. The electromagnetic force dominates over the gravitational force when $r<\mu_0^{1/2}e\hbar/G^{1/2}m^2\approx5000$ meters.
A: A fast answer: experimentally not feasible. Even n-n scattering itself is a huge problem: http://www.sbfisica.org.br/bjp/files/v35_850.pdf
The earlier answer about electromagnetic interaction seems reasonable (I upvoted), but still we miss the nuclear force part (what happens if EM force is switched off).
I think there is an important untold assumption of $r \rightarrow 0$ in the question, which would push the gravitational force to infinity.
If we assume two massive points on a typical nuclear distance : the order of $Gm^2/r^2$ is $10^{-11}10^{-27{^2}}/10^{-15^{2}} \sim 10^{-35}$ and when we try to speak about energy, we can arrive to the order of $10^{-50}$, which is to  me like $10^{-31}$ eV. Experimentally no way today.
Once we look at what is happening when two neutrons approach each other, we need a quantum mechanics. We use the scattering theory, where the system may be described as a sum of an incoming plane wave and a scattered outgoing wave (to all directions). This system is a version of nucleon-nucleon system and the energies here (e.g.bound state of proton-neutron) are typically in the order of $10^6$ eV. 
While two neutrons can interact in $S=0$ (mutual opposite spin positions) state, but they do not interact in $S=1$ (spins aligned) state, because it is excluded by the Pauli principle. And here - that are my feelings - we get to the last stage of the problem - quantum gravity. Does the gravitation work by exchanging a graviton between particles? Or what is the gravitational field of the particle described by a wave function? Does the gravity follow the Pauli principle as the other forces do? I dont know.
