We investigate collective multipole excitations for closed-shell nuclei from 16O to 208Pb by using correlated realistic nucleon-nucleon interactions in the framework of the random phase approximation (RPA). The dominant short-range central and tensor correlations are treated explicitly within the unitary correlation operator method (UCOM), which provides a phase-shift-equivalent correlated interaction VUCOM adapted to simple uncorrelated Hilbert spaces. The same unitary transformation that defines the correlated interaction is used to derive correlated transition operators. Using VUCOM, we solve the Hartree-Fock (HF) problem and employ the single-particle states as a starting point for the RPA. By construction, the UCOM-RPA is fully self-consistent, i.e., the same correlated nucleon-nucleon interaction is used in calculations of the HF ground state and in the residual RPA interaction. Consequently, the spurious state associated with the center-of-mass motion is properly removed, and the sum rules are exhausted within �3%. The UCOM-RPA scheme results in a collective character of giant monopole, dipole, and quadrupole resonances in closed-shell nuclei across the nuclear chart. For the isoscalar giant monopole resonance, the resonance energies are in agreement with experiment, hinting at a reasonable compressibility. However, in the 1- and 2+ channels the resonance energies are overestimated because of missing long-range correlations and three-body contributions.