In this paper, for Earth-orbiting spacecraft, by exploiting the equinoctial orbital elements, we develop a control technique that uses a constant specific thrust (i.e., acceleration) to eliminate the J2 perturbation-induced rotation of the perigee for inclined and equatorial orbits. Specifically, for constant disturbing accelerations, we average over one orbital period the rate of change of the equinoctial orbital elements to obtain expressions that relate the secular growth of the argument of the perigee and the longitude of the perigee to constant radial acceleration controls. These control accelerations are used to eliminate the J2 perturbation-induced secular growth of the argument of the perigee and the longitude of the perigee for inclined orbits and equatorial orbits, respectively. To justify the framework proposed in this paper, we also provide analytical expressions for the secular growth of all the equinoctial orbital elements caused by constant accelerations in the radial, in-plane, and out-of-plane directions. Simulation results are presented to illustrate the efficacy of the radial acceleration control in eliminating the rotation of the perigee.