Nope.
You're describing a Doppler Positioning System.
GPS uses...well...the Global Positioning System, that wants the receiver to "see" four satellites for a good fix, using timing delays between sending satellites. The sent time is embedded in the signals. They don't use frequency shifts due to motion.
From the NIST website:
"Multiple satellites beam signals at the speed of light toward Earth, and your device receives the signals at slightly different times, based on how far away each satellite is from your location. These times can be used to calculate the differences in distance from each satellite to determine your location on Earth."
Satellites beam signals at the speed of light toward Earth, and your device receives the signals at slightly different times, based on how far away each satellite is from your location. These times can be used to determine your location on Earth
www.nist.gov