Not exactly. The way physicists tried to reconcile Newtonian mechanics with the Maxwell equations was by imagining that the space between the Earth and the Sun (at least) is not a void, but that it is filled with an actual substance, the luminiferous ether. Light waves were posited to be mechanical waves in this substance, and the measured speed-of-light limit would have been a simple frame dragging effect: the moving train pulls the ether with it.
Of course, the Michelson-Morley experiment proved that even if this substance exists, it can't be dragged around, so frame dragging can't be an explanation for the constant speed of light, making the whole idea unnecessary. Future observations predicted by SR and GR would probably not have been compatible anyway.
But, getting back to your point, if this ether had existed and had been the medium through which light propagates and had been draggable by moving objects, then this would not have been a contradiction or even modification of Newtonian mechanics.
Of course, the Michelson-Morley experiment proved that even if this substance exists, it can't be dragged around, so frame dragging can't be an explanation for the constant speed of light, making the whole idea unnecessary. Future observations predicted by SR and GR would probably not have been compatible anyway.
But, getting back to your point, if this ether had existed and had been the medium through which light propagates and had been draggable by moving objects, then this would not have been a contradiction or even modification of Newtonian mechanics.