The velocity pulse contained in near-fault ground motions have a tremendous impact on dam safety. Previous studies have mainly focused on the response of dams under near-fault seismic records without considering the obliquely incident seismic waves. In this study, the structure–soil interaction (SSI) is taken into consideration, and the nonlinear behavior of a conventional concrete roller-compacted concrete (CC-RCC) gravity dam under near-fault pulse records and non-pulse records is investigated with consideration of the obliquely incident P waves. On the basis of the dam site conditions, three groups of near-fault pulse records are chosen, and three corresponding non-pulse records are fitted by their acceleration response spectra. Combining with the viscous-spring artificial boundary, the wave input method is proposed to transform the near-fault seismic records into the equivalent nodal forces at the boundary of the foundation. The concrete damaged plasticity model is used for the nonlinear analysis. The results show that the pulse ground motions are more destructive than the non-pulse motions. The nonlinear behavior of the dam varies with the incidence angle of P waves and generally reaches a maximum at 60° and 75°, the worst damage occurs at the interface between different materials of the dam, and the spatial variation of its damage is very obvious under near-fault seismic records with various incidence angles. Therefore, the effect of the angle of obliquely incident seismic waves and near-fault pulse effect should be considered comprehensively in the seismic analysis of dams.