The absorption spectra of transition-metal-doped alkali halide systems are distinguished by characteristic features identified with transitions between electron states localized on the impurity ion. The theoretical description of these features depends on a realistic treatment of the host crystal electronic structure, as well as on the treatment of the impurity ion. The success of recent calculations using the self-interaction-corrected local-spin-density (SIC-LSD) form of density-functional theory in determining the band structure of alkali halide crystals makes SIC-LSD a good candidate for use in the impurity problem. In this work SIC-LSD is applied to the NaCl:Cu+ impurity system, using an embedded-cluster technique explicitly including seven near-neighbor shells of the host crystal ions around the Cu+ site, allowing a full characterization of the low-lying impurity excited states. The calculated values for the fundamental 3d4s and 3d4p impurity-ion transitions are in close agreement with corresponding features of the NaCl:Cu+ absorption spectrum. We compare and contrast our SIC-LSD results with those of other recent calculations for NaCl:Cu+.