We construct simulated galaxy data sets based on the High Energy X-ray Probe (HEX-P) mission concept to demonstrate the significant advances in galaxy science that will be yielded by the HEX-P observatory. The combination of high spatial resolution imaging (<20 arcsec FWHM), broad spectral coverage (0.2–80 keV), and sensitivity superior to current facilities (e.g., XMM-Newton and NuSTAR) will enable HEX-P to detect hard (4–25 keV) X-ray emission from resolved point-source populations within ∼800 galaxies and integrated emission from ∼6,000 galaxies out to 100 Mpc. These galaxies cover wide ranges of galaxy types (e.g., normal, starburst, and passive galaxies) and properties (e.g., metallicities and star-formation histories). In such galaxies, HEX-P will: 1) provide unique information about X-ray binary populations, including accretor demographics (black hole and neutron stars), distributions of accretion states and state transition cadences; 2) place order-of-magnitude more stringent constraints on inverse Compton emission associated with particle acceleration in starburst environments; and 3) put into clear context the contributions from X-ray emitting populations to both ionizing the surrounding interstellar medium in low-metallicity galaxies and heating the intergalactic medium in the z > 8 Universe.