We propose a new precise point positioning (PPP) method for a real-time low earth orbit (LEO) satellite positioning problem. Our approach uses the GPS navigation (GPS NAV) solution onboard a satellite as a satellite reference trajectory and estimates the position errors in real-time using the PPP technique. The dynamics of the errors is modeled by the first-order Gauss-Markov process and the GPS ionosphere-free carrier phase measurement model is sequentially linearized around the positions of the GPS NAV. The continuous-discrete Kalman filter is applied to the problem of interest for the state estimation. The advantage of our approach is that the calculation cost is much less demanding compared with approaches that integrate the satellite precise equation of motion. Moreover, as the accuracy of the GPS NAV improves, the accuracy of the proposed method also increases and we expect that happens after the full operation of the GPS III. The experimental evaluation using the Japanese ALOS2 (Advanced Land Observing Satellite), which is a LEO radar earth observation satellite, showed the effectiveness of our approach: 60 to 80 centimeters accuracy improvement on average over that for the GPS NAV whose accuracy was about 2 meters.