The ponderomotive stabilization of plasma in an axisymmetric mirror device is considered. It is assumed that (i) the RF field is excited by a full-turn-loop antenna and (ii) its frequency is below the eigenfrequency spectrum of the plasma column. It is shown that the dissipation of the RF power drives an instability that originates from a resonant-particle interaction. This instability is weak except for the case when the minority ion scheme of ICRF heating is used. However, even in the latter case, the resonant-particle instability can be suppressed by quasilinear diffusion of the minority ions in the velocity space at a sufficiently high level of RF power. With the instability suppressed, a simple criterion for the ponderomotive force to stabilize flute perturbations in mirror plasmas is obtained.