Generalized-Kohn-Sham (GKS) orbital energies obtained self-consistently from the random phase approximation energy functional with a semicanonical projection (spRPA) were recently shown to rival the accuracy of GW quasiparticle energies for valence ionization potentials. Here, we extend the scope of GKS-spRPA correlated one-particle energies from frontier-orbital ionization to core orbital ionization energies, which are notoriously difficult for GW and other response methods due to strong orbital relaxation effects. For a benchmark consisting of 23 1s core electron binding energies (CEBEs) of second-row elements, chemical shifts estimated from GKS-spRPA one-particle energies yield mean absolute deviations from experiment of 0.2 eV, which are significantly more accurate than the standard GW and comparable to Δ self-consistent field theory without semiempirical adjustment of the energy functional. For small ammonia clusters and cytosine tautomers, GKS-spRPA based chemical shifts capture subtle variations in covalent and noncovalent bonding environments; GKS-spRPA 1s CEBEs for these systems agree with equation-of-motion coupled cluster singles and doubles and ADC(4) results within 0.2–0.3 eV. Two perturbative approximations to GKS-spRPA orbital energies, which reduce the scaling from O(N6) to O(N5) and O(N4), are introduced and tested. We illustrate the application of GKS-spRPA orbital energies to larger systems by using oxygen 1s CEBEs to probe solvation and packing effects in condensed phases of water. GKS-spRPA predicts a lowering of the oxygen 1s CEBE of approximately 1.6–1.7 eV in solid and liquid phases, consistent with liquid-jet X-ray photoelectron spectroscopy and gas phase cluster experiments. The results are rationalized by partitioning GKS-spRPA electron binding energies into static, relaxation, and correlation parts.