The United States satellite navigation system that relies on relativistic corrections is GPS, the Global Positioning System. Its satellites carry atomic clocks whose timing must be adjusted because both motion and gravity affect the passage of time.
Special relativity makes a moving satellite clock run slightly slower than a clock on Earth. General relativity produces the opposite effect: because a GPS satellite is farther from Earth's gravitational field, its clock runs faster. The gravitational effect is larger, so the combined result is that a satellite clock gains about 38 microseconds per day relative to a clock on the ground.
That difference sounds tiny, but GPS determines position by measuring how long radio signals take to travel from satellites to a receiver. Without relativistic corrections, position errors would accumulate at roughly 10 kilometers per day. Engineers compensate by adjusting the satellite clock frequency before launch and by applying additional corrections during navigation calculations.
BeiDou, Galileo, and GLONASS are also satellite navigation systems and must account for relativity too. GPS is the correct answer here because the question asks specifically for the United States system.