How far away does a neutral atom have to be from another neutral atom before it obeys gravity?

Mass (such as a neutral atom) always obeys gravity. It's not clear to me what you're asking.

EDIT:
If they're neutral, then the EM force is zero and gravity is stronger.

If they're not neutral then it depends on the relative mass and charges, but in general gravity is weaker than the other forces by several orders of magnitude.

As an example, let's compare the attraction and repulsion of two protons.

The attractive force of gravity is given by:
F(g) = GMm/d^2. So at a distance of 1cm, the gravitation attraction would be:

F(g) = (6.67x10^-11 N m^2/kg^2) (1.67x10^-26kg)^2 / (0.01m)^2
F(g) = 1.87 x 10^-60 N

The repulsive force from the identical EM is given by Coulomb's law
F(e) = 1/(4 pi epsilon0) q1 q2 / d^2
F(e) = 1/(4 pi 8.85x10^-12 A^2 s^4/kg m^3) (1.60x10^-19 A s)^2 / (0.01m)^2
F(e) = 2.31 x 10^-24N

F(g) / F(e) = 1.87x10^-60N / 2.3x10^-24N
F(g) / F(e) ~ 10^-36

So in this case it's around 36 orders of magnitude stronger.
lets say without external forces and at earth's surface,