Bzzzt! No, when there is no movement, there is zero relative velocity. If there was no acceleration you would be weightless. Since you aren't weightless when you hang suspended, or stand "still", you must NOT have zero acceleration. Therefore you must be accelerating, therefore gravity accelerates mass. G is a constant, g is only a constant locally. When you "eliminate F = mg", what are you actually doing? Where does it "go"?