Hard to say since there is practically no theoretical basis that I know of suggesting any way to develop FTL travel. It will be difficult to get to other systems though without something FTL, and FTL generally means something extra-(as in "beyond")-Newtonian.
Some Sci-Fi writers who avoid considerations of Newtonian movement are doing so because they want to simplify things and make them more recognizable and/or appealing to unsophisticated audiences, or to enable or disable certain kinds of situations. Other writers are themselves more interested in a certain imaginary existence than they are about making probable predictions.
I don't think Newtonian movement or conservation of momentum will ever "go away" though, as some bad sci fi flick animation seems to suggest.
PvK
Re: Newtonian ships or not?.
Threaded Mode