We study relativistic effects in photoionization of a hydrogen atom exposed to an intense laser pulse of general ellipticity. The frequency of the laser pulse resides in the ultraviolet region. To this end, the semirelativistic approach introduced in Kjellsson Lindblom et al. [T. Kjellsson Lindblom, M. Forre, E. Lindroth, and S. Selsto, Phys. Rev. Lett. 121, 253202 (2018)] is applied. We present in some detail how this approximation is derived from the Dirac equation for elliptically polarized light within the so-called long-wavelength approximation. The validity of the semirelativistic approach is confirmed by direct comparison with the solution of the Dirac equation. It is found that the total ionization yield depends very weakly on ellipticity in the case of ionization from the isotropic ground state. With the excited initial state n = 2, l = m(l) = 1; however, pronounced ellipticity dependence is seen-in particular at the stabilization peak. Albeit small, relativistic corrections to the ionization probabilities are found. The correction is found to be largest for linear polarization. While relativistic effects tend to reduce the total ionization probability for most intensities considered, we also report a slight relativistic enhancement at comparatively modest field strengths.