Does GPS use the theory of relativity? Yes.
The information below is quoted from GPS interface specification.
SV here is Space Vehicle. This is how the frequency of the clock set:
The nominal frequency of this source - as it appears to an observer
on the ground - is 10.23 MHz. The SV carrier frequency and clock
rates - as they would appear to an observer located in the SV - are
offset to compensate for relativistic effects. The clock rates are
offset by ∆f/f = -4.4647E-10, equivalent to a change in the P-code
chipping rate of 10.23 MHz offset by a ∆f = -4.5674E-3 Hz. This is
equal to 10.22999999543 MHz.
These numbers are explained in detail here.
This is not the only clock correction, there are many effects discussed in the paper, and so:
[SV message] ... contain the parameters needed by the users for apparent SV clock correction
(toc, af2, af1, af0). The related algorithm is given in paragraph
The algorithms defined below (a)
allow all users to correct the code phase time received from the SV
with respect to both SV code phase offset and relativistic effects
SV basically keeps track of its clock's error which is calculated by the communication with the Control Segment (CS):
The NAV data contains the requisite data for relating GPS time to UTC.
The accuracy of this data during the transmission interval shall be
such that it shall relate GPS time (maintained by the MCS of the CS)
to UTC (USNO) within 90 nanoseconds (one sigma).
Also, it is outlined how the user should calculate the relativistic shifts given the data from SV:
the user's equipment must determine the requisite relativistic
correction. Accordingly, the offset given below includes a term to
perform this function...
There is a list of different generations of GPS blocks which have different periods in which they can function independently from Control Segment. If no data (including time correction) has been uploaded to the satellite within this period, it's health is considered "bad" and it is no longer used (until revived).
Each Block IIR/IIR-M/IIF SV in the constellation determines its
own ephemeris and clock correction parameters via SV-to-SV ranging,
communication of data, and on-board data processing which updates data
uploaded by the CS. In the Autonav mode the Block IIR/IIR-M/IIF SV
will maintain normal operations as defined in paragraph 22.214.171.124 In
the Autonav mode the Block IIR/IIR-M/IIF SV will maintain normal
operations as defined in paragraph 126.96.36.199 precision. If the CS is
unable to upload the SVs, the Block IIR/IIR-M/IIF SVs will maintain
normal operations for period of at least 60 days after the last
The document is huge and there's lots of interesting information in there. One of the things to derive from it is that the newer generations of GPS devices take into account more time correction effects with higher precision and thus require less frequent calibration from Control Segment.
Does GPS require the knowledge of relativity? No.
The GR correction can be done without the knowledge of where it comes from. The signal frequency received from a satellite is different from the frequency originally emitted, and this shift can be measured experimentally. Knowing this shift, then one can proceed with the whole procedure described above, and all the other minor relativistic corrections would be included in on-line calibration, exactly as such hard-to-estimate or random effects are included now, as described here:
These updates synchronize the atomic clocks on board the satellites to
within a few nanoseconds of each other, and adjust the ephemeris of
each satellite's internal orbital model.
This is not only possible, but exactly how it is done at the moment, because there are a lot of factors contributing to the inaccuracies and to calibrate the device you need to know the adjustments exactly.
So, would it be possible to create GPS system without the knowledge of Theory of Relativity? Yes, as described above, using experimental results, calibration and trial and error method. However, it would suggest the presence of unexplained phenomenon(a), because the frequency shift would not be explainable by the known phenomena. It likely will not work very well as it will require more frequent clock correction (I would not speculate further because it is too hard to guess what "would be" in such case).
Does GPS prove Relativity (General and/or Special)? Yes and no.
All I could find is a minor claim that the satellite clocks initial
setting were CONSISTENT with relativity
A scientific theory is a well-substantiated explanation of some aspect
of the natural world that is acquired through the scientific method,
and repeatedly confirmed through observation and experimentation.
There is no other way to confirm a theory but by observation. A consistent confirmation of a theory through observation constitutes a proof of a theory. The emitted frequency from the satellite is different from the frequency observed by the receiver, and the difference closely matches the GR time shift. In this instance, the theory matches (and successfully predicts) the observation. Of course, GPS alone is not a proof (in the aforementioned sense) of GR and SR, no serious scientific resource claims this. This is just one of the many demonstrations of the predictive power of these theories and there are certainly multiple other experiments and observations which provide a much more elaborate proof for both of these.