The complex phase method has been further extended to the problem of electromagnetic (EM) field scintillations on Earth-satellite GPS paths of propagation. The numerical and analytic technique based on the method has been developed to characterize the transionospheric channel of propagation. The effects of additional range errors due to the ionospheric electron density fluctuations in space and time have been studied taking into account the ray bending due to the inhomogeneous background ionosphere and the diffraction on local random ionospheric inhomogeneities. In the method developed, the impact of the Earth's magnetic field is accounted for by the anisotropic spatial spectrum of the ionospheric turbulence with different outer scales along and across the magnetic field lines. The variances of the EM field phase (yielding range errors) and level (log amplitude) fluctuations have been calculated for different models of the background ionospheres characterized by different height electron density profiles and total electron content. The conditions of the saturated regime of propagation, which will likely result in the degradation of a GPS navigation system, have been discussed. In addition, the scattering function of the GPS transionospheric channel of propagation has been constructed and simulated for a wideband signal.