The residual stress in Cu films epitaxially grown on GaAs(001) single crystalline substrates has been compared to polycrystalline Cu growth on (111) textured Au substrates, both grown via the same galvanostatic electrodeposition process. Optimized Cu/GaAs epitaxial nucleation and growth was obtained with a substrate pre-etch in dilute ammonium-hydroxide followed by electrodeposition in a pure Cu sulfate aqueous electrolyte at elevated temperatures. The resulting films are single crystalline, and strain relaxed, as measured by X-ray and electron diffraction. The (001) surfaces developed square pyramidal facets that increase in average size with increasing current density. In situ wafer curvature measurements found that the Cu/GaAs films followed the commonly observed change from compressive to tensile and finally to compressive stress (-200 MPa) that quickly relaxed once growth was interrupted. In contrast, polycrystalline Cu/Au films developed a smaller and constant tensile stress (+10 MPa) that relaxed more slowly. Given the similar growth rates of the two systems, differences in residual stress are related to differences in the density and nature of the coalescence boundaries and associated surface adatom processes. The resulting electrical properties of Cu/GaAs diodes show an interfacial capacitance that is consistent with interdiffusion and the reaction layer detected by electron microscopy.