In this paper, we propose and analyze a globally convergent regularized Newton method with positive definite regularization for solving nonsmooth optimization problems. Our approach leverages the coderivative-generated second-order subdifferential (generalized Hessian) and replaces the identity matrix in traditional algorithms with a general positive-definite symmetric matrix to regularize the generalized Hessian. By appropriately selecting the regularization matrix, we enhance the practical performance of the algorithm. Under suitable assumptions, we establish the well-posedness of the proposed algorithm. Using tools from variational analysis and generalized differentiation, we derive explicit convergence rates under the Holder strong metric subregularity condition. Specifically, we quantify the precise relationship between the algorithm’s convergence rate and the order of Holder strong metric subregularity. For a class of nonsmooth functions, namely prox-regular functions, corresponding algorithms have also been developed via their Moreau envelopes. As an application, we apply the proposed method to convex composite optimization problems within the forward-backward envelope framework. Numerical experiments on Lasso problems demonstrate that our algorithm outperforms some existing methods.