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?

EDIT : @Ruslan answer gives the drawbacks of putting the $\gamma$ factor in the equation of Gravitoelectromagentism and it is very fascinating that this concept was introduced by Oliver Heaviside much before Einstein's GR. But I want to know the drawbacks of putting $\gamma$ factor in Newtonian Gravity,the gravity which is governed totally by mass. Why Newtonian Gravitational Equation can't be modified by introducing $\gamma$ factor? Is the non-invariance the only problem (I know it's a big problem but is it the only one)?

A Side Note:- Due to some reason people are thinking that my question is about the concept of relativistic mass, the concept to which many links are already present. I want to clarify that my question is not: How to interpret $m=\frac{m_0}{\sqrt{1-v^2/c^2}}$ ?. But my question is why a direct substitution of this formula in Newtonian Gravity equation (as @QMechanics has said in his answer) would lead to contradictions and what are those contradictions?

@Ruslan answers about the drawbacks of GEM, his answer focuses as to why GEM needs to be modified even further for strong fields. My question is why did Newtonian Gravity failed in relativistic mechanics? Why GEM was introduced, was it only for invariance principle or were the predictions of introducing $\gamma$ factor in Newtonian Gravity very inconsistent?. I'm talking about introducing $\gamma$ factor because of it's advent in the momentum equation, my question is not just a random thought. To restate my question, I would write :

What are the illogical predictions of Newtonian Gravity if we introduce the $\gamma$ factor with mass?