The atomic and electronic band structures of GaP/Si(001) heterointerfaces were investigated by $ab\; initio$ density functional theory calculations. Relative total energies of abrupt interfaces and mixed interfaces with Si substitutional sites within a few GaP layers were derived. It was found that Si diffusion into GaP layers above the first interface layer is energetically unfavorable. An interface with Si/Ga substitution sites in the first layer above the Si substrate is energetically the most stable one in thermodynamic equilibrium. The electronic band structure of the epitaxial GaP/Si(001) heterostructure terminated by the (2×2) surface reconstruction consists of surface and interface electronic states in the common band gap of two semiconductors. The dispersion of the states is anisotropic and differs for the abrupt Si-Ga, Si-P, and mixed interfaces. Ga 2$p$, P 2$p$, and Si 2$p$ core-level binding-energy shifts were computed for the abrupt and the lowest-energy heterointerface structures. Negative and positive core-level shifts due to heterovalent bonds at the interface are predicted for the abrupt Si-Ga and Si-P interfaces, respectively. The distinct features in the heterointerface electronic structure and in the core-level shifts open new perspectives in the experimental characterization of buried polar-on-nonpolar semiconductor heterointerfaces.