As I understand it, the notion of falsifiability is really more like, "Theory X is acceptable if it can be shown to be false by the use of tools whose principles meet the same standard of falsifiability, even if those tools aren't currently available."
So a theory that can be tested only with later scientific refinements -- say, by increasing measurement precision beyond what's currently available -- is indeed eligible to be classified as science. That criterion would allow for Newtonian mechanics to be accepted in its time, and for things like string theory to be accepted provisionally in ours.
Basically, the ultimate failure of the Newtonian model can't be used as an argument that it should never have been considered valid science.
So a theory that can be tested only with later scientific refinements -- say, by increasing measurement precision beyond what's currently available -- is indeed eligible to be classified as science. That criterion would allow for Newtonian mechanics to be accepted in its time, and for things like string theory to be accepted provisionally in ours.
Basically, the ultimate failure of the Newtonian model can't be used as an argument that it should never have been considered valid science.