In the Bohr atom, the negative electron and positive proton are attracted by the electrostatic force between them. Electrons in the earth are also attracted to protons in the moon, but they are equally repelled by the similar number of electrons in the approximately neutral moon. The effect is an overall cancellation. Because of this typically neutral state of massive objects in the universe, the electrostatic force plays no role at this scale.
Just as the electron and proton in the nucleus are electrically attracted, they are also gravitationally attracted. But if you'll do the calculations, you'll see that the gravitational force is negligibly small in comparison to the electrostatic force.