Wear is a tremendously important phenomenon, which takes place on the surfaces of two solids in contact under cyclic loads and constitutes one of the most-significant ways of failure for mechanical elements. However, it is not the only source of failure in contacting solids. The subsurface stresses should also be considered, due to the fatigue and crack initiation problems. Nevertheless, these stresses (i.e., their maximum values and distributions) evolve with the solids’ surface wear (i.e., with the load cycles) and also depend on the friction intensity. Therefore, their evolution should be properly computed to predict failures in mechanical elements under wear conditions. This work focused on the study of the evolution of the surface wear and the subsurface stress distributions generated—in an elastic half-space—by a cylindrical flat-ended punch, under cyclic indentation loading (i.e., radial fretting wear conditions). Based on a numerical scheme recently presented by the authors, this is the first time that, for this contact problem, the surface wear and subsurface stress distribution (i.e., maximum value and its location)—and its evolution—were simultaneously analyzed when orthotropic friction and fretting wear conditions were considered. The studies presented in this work were developed for purely elastic contact assumptions.