We realize the growth of self-catalyzed GaAsP nanowires (NWs) on silicon (111) substrates using solid-source molecular beam epitaxy. By optimizing the V/III and P/As flux ratios, as well as the Ga flux, high-crystal-quality GaAsP NWs have been demonstrated with almost pure zinc-blende phase. Comparing the growth of GaAsP NWs with that of the conventional GaAs NWs indicates that the incorporation of P has significant effects on catalyst nucleation energy, and hence the nanowire morphology and crystal quality. In addition, the incorporation ratio of P/As between vapor-liquid-solid NW growth and the vapor-solid thin film growth has been compared, and the difference between these two growth modes is explained through growth kinetics. The vapor-solid epitaxial growth of radial GaAsP shell on core GaAsP NWs is further demonstrated with room-temperature emission at ~710 nm. These results give valuable new information into the NW nucleation mechanisms and open up new perspectives for integrating III-V nanowire photovoltaics and visible light emitters on a silicon platform by using self-catalyzed GaAsP core-shell nanowires.