According to special theory of relativity, we know that any equation that contains mass is corrected by $$ m\to \frac{m_0}{\sqrt{1-v^2/c^2}},$$

where $m_0$ is the rest mass. In the case of momentum, the classical momentum i.e. $mv$ we modify it by writing $$ p = \frac{m_0v}{\sqrt{1-v^2/c^2}}.$$

And since Newton's second law says "Force is the rate of change of momentum", we can write it as $$ F = m_0 \frac{d}{dt}\left(\frac{v}{\sqrt{1-v^2/c^2}}\right).$$

So far so good, now let's turn to Newton's law of universal gravitation which says that gravitational field produced my mass $m$ is $$ g= \frac{Gm}{r^2}.$$

Now, why can't we modify it by writing $m$ out like $$ g = \frac{Gm_0}{r^2\sqrt{1-v^2/c^2}}?$$ Is there any physical violation in writing this out or it is just because of general relativity which doesn't consider gravity as force but as consequence of curved space-time?