We have used inelastic neutron scattering measurements to study the magnetic excitations of Co core/CoO shell nanoparticles for energies from 0-50 meV. Above the blocking temperature TB, broad quasielastic scattering is observed, corresponding to the reorientation of the Co core moments and to paramagnetic CoO scattering. Below TB, two nearly dispersionless inelastic peaks are found, whose energies increase with decreasing temperature as order parameters, controlled by the nanoparticle Néel temperature TN =235 K, and saturating as T→0 at 2.7 meV and 6.7 meV, respectively. Similar excitations were observed in a powdered single crystal of CoO, indicating that both are intrinsic excitations of CoO, resulting from the exchange splitting of single ion states for T≤TN . Pronounced finite size effects are observed for the scattering from the CoO nanoparticle shells, whose thicknesses range from 1.7-4.5 nm. These include an enhanced excitation linewidth, as well as a response that is not only spread over a much wider range of wave vectors, but is also significantly more intense in the nanoparticles than in bulk CoO.