Newton's gravity depends on the euclidean distance between two masses. Two frames moving with respect to each other will have different values of length between masses so the forces will be different in two frames. Is it enough to prove that the gravity rule has to be modified?
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Yes. Your affirmation is essentially that Newtonian gravity is not Lorentz invariant. However, it is certainly not enough to say how the modification should be done. For example, the same could be said for Electrostatics, and one would arrive at the conclusion that Electrostatics is not enough: one needs Electrodynamics. Gravity, on the other hand, needs GR (one can describe weak gravitational fields in terms of gravitoelectromagnetism, but it leads to runaway solutions (see this post) and hence can't be a fundamental description).
In short, your argument is sufficient to say there is something more besides what is described by Poisson's equation in Newtonian gravity. However, it is not enough to say what is lacking.
Níck Aguiar Alves
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