Depends on what one calls "the space program." The Apollo and Space Shuttle programs had jack squat to do with GPS.
Now, of course, GPS is a "space program" since it requires satellites, but the whole program was part of the military unmanned program. And the launchers for all the satellites are ancient early 60s technology, basically.
Actually, the orbital science done during Gemini and Apollo and some side experiments with spacecraft tracking had a lot to do with proving that GPS would work at all, and thus get it financed. I don't doubt that had we quit with the building of the ICBM fleet, GPS wouldn't exist.
One of the lead computer navigation people on Apollo was named "Garmin". I haven't found out yet if he was "The" Garmin, but I wouldn't doubt it.