In this paper, we propose a generalized Moreau enhanced minimization induced (GME-MI) regularization model and its proximal splitting algorithm for further improvement of the MI penalty derived as the minimum of a convex function. We first design the GME-MI penalty function by applying the GME construction to the MI penalty, and derive an overall convexity condition for the GME-MI regularized least-squares model. Then, under the overall convexity condition, characterizing the solution set of the GME-MI model with a carefully designed averaged nonexpansive operator, we develop a proximal splitting algorithm which is guaranteed to converge to a globally optimal solution. Numerical examples demonstrate the effectiveness of the proposed approach.