The brief answer is no.
The gravitational force between protons in a nucleus is miniscule , about 2 x10-36N, or about 1 ten billionth of the weight of a proton. The electrostatic (Coulomb) force between protons is about 2 N, i.e. about the weight of 1026Protons. This repulsive force is exceeded by the attraction of the strong nuclear force, which is why all nuclei don't fly apart due to electrostatic repulsion. The forces on an electron in an inner shell due the a proton in the nucleus are - gravitation about 1x10-47 N (about 1 billionth of a billionth of the weight of an electron) and the electrostatic force between an electron and a nuclear proton is 2x10-8 N, which is the weight of about 2x1021 electrons.
So not only will the electrostatic force not shield objects from gravitation (otherwise objects inside of charged containers experience a reduced gravity), but the gravitational forces are negligible compared to the electrostatic forces between the components of the atom and the nucleus and the nuclear components experience the Strong nuclear force which is even stronger than that.
Current understanding is that a free proton (i.e. one in space outside of the nucleus) is perfectly stable - lifetimes greater than the universe lifetime. Neutrons in the nucleus are stable, but free neutrons have a lifetime of only about 14 minutes, it is believed that the nuclear neutrons are stabilized but the Strong Nuclear force., but I don't know how that happens.