A fully self-consistent renormalized random-phase approximation is constructed based on the self-consistent Hartree-Fock mean field plus exact pairing solutions (EP). This approach exactly conserves the particle number and restores the energy-weighted sum rule, which is violated in the conventional renormalized particle-hole random-phase approximation for a given multipolarity. The numerical calculations are carried out for several light-, medium-, and heavy-mass nuclei such as 22O, 60Ni, and 90Zr by using the effective MSk3 interaction. To study the pygmy dipole resonance (PDR), the calculations are also performed for the two light and neutron-rich 24,28O isotopes, whose PDRs are known to be dominant. The results obtained show that the inclusion of ground-state correlations beyond the random-phase approximation (RPA) by means of the occupation numbers obtained from the EP affects the RPA solutions within the whole mass range, although this effect decreases with increasing the mass number. At the same time, the antipairing effect is observed via a significant reduction of pairing in neutron-rich nuclei. The enhancement of PDR is found in most neutron-rich nuclei under consideration within our method.