Quadrature compressive sampling (QuadCS) is a recently introduced sub-Nyquist sampling scheme for effective acquisition of inphase and quadrature (I/Q) components of sparse radio frequency signals. In applications to pulse-Doppler radars, the QuadCS outputs can be arranged into a two-dimensional data format, in terms of slow time and virtual fast time, similar to that by Nyquist sampling. This paper develops a compressive sampling pulse-Doppler (CoSaPD) processing scheme which performs Doppler estimation/detection and range estimation from the sub-Nyquist data without recovering the Nyquist samples. The Doppler estimation is realized through a spectrum analyzer as in classical processing, whereas the detection is performed using the Doppler bin data. The range estimation is performed using sparse recovery algorithms only for the detected targets to reduce the computational load. A low detection threshold is used to improve the detection probability and the introduced false targets are then removed in the range estimation stage by exploiting the inherent target detection capability of the recovery algorithms. Simulation results verify the effectiveness of the proposed CoSaPD scheme, which requires only one-eighth of the Nyquist rate to achieve similar performance to the classical processing with Nyquist samples, provided that the input signal-to-noise ratio (SNR) is above -25 dB.