We present structural characterization of the wurtzite crystal structure of GaP nanowires, which were recently shown to have a direct electronic band gap. The structural parameters of the wurtzite phase do consist of two lattice parameters and one internal degree of freedom, determining the Ga-P bond length along the c direction. Using density functional theory calculations, we study the influence of the internal degree of freedom on the band structure. By synchrotron x-ray diffraction studies near the Ga-K edge we determine the lattice parameters a = 3.841 A and c = 6.335 A as well as the internal degree of freedom u = 0.37385 with high accuracy. We find that different Ga-P bond lengths are not equal, in contrast to the case in the zinc blende bulk phase. As a result, a spontaneous polarization is predicted for wurtzite GaP.