You absolutely cannot make it work without relativity being taken into account. Why is this even a question?
Because an observer on the ground sees the satellites in motion relative to them, Special Relativity predicts that we should see their clocks ticking more slowly (see the Special Relativity lecture). Special Relativity predicts that the on-board atomic clocks on the satellites should fall behind clocks on the ground by about 7 microseconds per day because of the slower ticking rate due to the time dilation effect of their relative motion [2].
Further, the satellites are in orbits high above the Earth, where the curvature of spacetime due to the Earth’s mass is less than it is at the Earth’s surface. A prediction of General Relativity is that clocks closer to a massive object will seem to tick more slowly than those located further away (see the Black Holes lecture). As such, when viewed from the surface of the Earth, the clocks on the satellites appear to be ticking faster than identical clocks on the ground. A calculation using General Relativity predicts that the clocks in each GPS satellite should get ahead of ground-based clocks by 45 microseconds per day.
The combination of these two relativistic effects means that the clocks on-board each satellite should tick faster than identical clocks on the ground by about 38 microseconds per day (45-7=38)! This sounds small, but the high-precision required of the GPS system requires nanosecond accuracy, and 38 microseconds is 38,000 nanoseconds. If these effects were not properly taken into account, a navigational fix based on the GPS constellation would be false after only 2 minutes, and errors in global positions would continue to accumulate at a rate of about 10 kilometers each day! The whole system would be utterly worthless for navigation in a very short time.
This is not hard to find out. The confusion I think arises because (most or all?) of the satellites and GPS itself doesn’t actually use Einstein’s field equations to compensate for relativistic effects because it would be too complex and not worth it. Instead, the GPS systems use clock corrections and various other hacks to fudge something “good enough” to avoid having to properly deal with relativistic effects. See here for more details (a bit outdated but still relevant).
So, as usual, the world is more complex than most people want to deal with. But in short, yes, GPS needs to compensate for relativistic effects to work correctly. No, it doesn’t actually use Einstein’s equations but again, yes, relativity matters here very much.
Tired of doing other people’s thinking for them, but I am better at it than they are, so….