The magnetic-field component of a very intense laser field interacting with an atom cannot normally be neglected for intensities that lead to relativistic velocities of the electrons. Here we investigate stabilization and harmonic generation in this relativistic regime from a three-dimensional hydrogen atom modeled as a classical system with a distribution of initial conditions derived from a Monte Carlo average. Particular emphasis is placed on the problems of ionization in the direction of propagation of the applied laser field, which will be shown to arise from the inclusion of the magnetic-field component of the laser field. In the harmonic spectra, Doppler shifting occurs for observation directions orthogonal to the direction of laser-field propagation. Retardation effects show up in the harmonic spectra in the forward direction and inhibit the magnetic-field effects of the free-electron contribution of the forward-direction spectrum. In general, few harmonics are observed in our single-atom treatment because of the magnetically induced three-dimensional motion of the electron for intensities approaching the relativistic regime, and because of high ionization probabilities, i.e., also the breakdown of stabilization, for strongly relativistic laser intensities.