Gravitational field strength Can I use $g=GM/r^2$ to calculate the gravitational field strength proton or electron or any other particles? If not then why? If yes then what would be that really mean?
 A: You could use Newton's classic equation, but, as illustrated in an answer here, its effects would be almost negligible.
Aside from that, we don't know if Newtonian gravity even applies to particles on that scale. To answer that question would require a theory of quantum gravity which, to date, has not yet been developed.
A: Usually (that I know of) the noticeable differences between General Relativity and Newtonian gravity only become apparent on a macroscopic, $>=$ planetary scale.  For example the precession of Mercurcy, light bending, black holes etc...
So for atomic scales, you can use the Newtonian formula, yes. 
It tells you the strength of the gravitational field generated by, say, a proton of mass m. So if you place a test mass of mass $m_1$, it will experience a force towards the proton $F = m_1g$.  Due to the low masses and the small value of $G$, gravitational effects on atoms are almost always superseded by the electromagnetic interaction - i.e. a proton and an electron will form an EM-bound state before they would form a gravitational bound state (an orbit), or simply scatter from one another.
