Often, when there are question on why fundamental research makes sense or is neccesary, I've seen answers like "GPS is used everywhere today, and it wouldn't work without relativity".
Of course, if we didn't know about relativity - basically, if we still assumed physics was newtonian - and built a GPS system around this assumption, it wouldn't work in the "real world". We'd be terribly puzzled why those clocks that worked so well on earth suddenly ran faster on a satellite.
However, assuming we had a newtonian universe, with the speed of light being finite, and clocks running at the same speed no matter how far down a gravity well they are - is there any reason why we wouldn't be able to build a GPS system? Except of course, that that kind of universe wouldn't work at all so the human species couldn't have developed?
What the question boils down to is, I guess - is the fact that our universe is relativistic that's needed to build a GPS system, or is it just the fact that we know about relativity?
Please notice that this isn't a duplicate of Why does GPS depend on relativity?; that question asks why we need to correct for relativistic effects, while I'm asking if/why GPS isn't able to work in a world that doesn't have them.