I've seen that often geometric units are defined by setting $G=c=1$, however, I have been working with a different definition, namely, to multiply time in seconds by $c$ so that it's measured in meters, and multiply the mass in kilograms by $\frac{G}{c^2}$ so that it's measured in meters.
It should then follow that the law of universal gravitation expressed in geometric units is rewritten as $$F=\frac{Mm}{r^2}$$ But I can't seem to get this expresion. Consider the original one $$F=G\frac{Mm}{r^2}$$ Since force is measured in $kg\cdot m\cdot s^{-2}$, we need to multiply the left side by $\frac{G}{c^2}\frac{1}{c^2}$ and we need to multiply the right side by $\frac{G^2}{c^4}$ since there is two masses. The equation the results $$F\frac{G}{c^4}=G\frac{Mm}{r^2}\frac{G^2}{c^4}\iff F=G^2\frac{Mm}{r^2}$$ As you can see, I don't get the desired result, that is, that $G$ dissapears (or is equal to $1$).
EDIT: I just noticed the mistake and posted it as an answer to this question.