This paper introduces a new policy-regularized model-predictive control (PR-MPC) approach to automatically generate and stabilize a diverse set of quadrupedal gaits. Model-predictive methods offer great promise to address balance in dynamic robots, yet require the solution of challenging nonlinear optimization problems when applied to legged systems. The new proposed PR-MPC approach aims to improve the conditioning of these problems by adding regularization based on heuristic reference policies. With this approach, a unified MPC formulation is shown to generate and stabilize trotting, bounding, and galloping without retuning any cost-function parameters. Intuitively, the added regularization biases the solution of the MPC towards common heuristics from the literature that are based on simple physics. Simulation results show that PR-MPC improves the computation time and closed-loop outcomes of applying MPC to stabilize quadrupedal gaits.