A crystal plasticity finite element method constitutive model was developed to investigate the anisotropic mechanical behaviors of (001), (011), and (111) initially orientated copper (Cu) single crystals during nanoindentation deformation. The numerical load-indentation depth curve and hardness-indentation depth curve were compared with experimental observations to validate the established three-dimensional (3D) CPFEM model. The difference of indentation loads between (111) crystal and (001) crystal is ~10.68 pct, and the difference of indentation modulus between (111) surface and (001) surface is ~10.80 pct. The numerical results show the noticeable indentation size effect for three crystals, and slightly different hardness values on different crystallographic planes. The equivalent plastic strain and lattice rotation angles were also analyzed on three through-thickness cross sections to study the plastic deformation-induced texture anisotropy.