  The proton inelastic scattering of $^{24}$O($p,p'$) at 62 MeV/nucleon is
described by a self-consistent microscopic calculation with the continuum
particle-vibration coupling (cPVC) method. The SLy5, SkM*, and SGII parameters
are adopted as an effective nucleon-nucleon interaction. For all the
parameters, the cPVC calculation reproduces very well the first peak at 4.65
MeV in the $^{24}$O excitation energy spectrum as well as its angular
distribution. The role of the cPVC self-energy strongly depends on the
effective interactions. The higher-lying strength around 7.3 MeV is suggested
to be a superposition of the $3^-$ and $4^+$ states by the results with SLy5
and SGII, whereas the SkM* calculation indicates it is a pure $3^-$ state. This
difference gives a rather strong interaction dependence of the angular
distribution corresponding to the higher-lying strength.
