Once a theory or even a law becomes unfalsifiable its not longer science. Until every observation has been made, it remains possible a contradiction will be discovered. Therefore nothing can ever be settled.
Your conclusion doesn't follow. Where does it say that for something to be settled, it must unfalsifiable. For instance, I think common understanding is that "settled science" is merely the collection of theories which are widely accepted and very unlikely to be proven false.
With that said there are lots of cases like inertia where the evidence in support of it is so strong and so complete; we can reasonably depend upon its truthfulness and pretty much reject anyone who disputes it unless they have some really really solid independently reproducible observations to the contrary.
That would be the "settled science" we're talking about. It's settled, not indestructible.