Stability certification and identification of a safe and stabilizing initial set are two important concerns in ensuring operational safety, stability, and robustness of dynamical systems. With the advent of machine-learning tools, these issues need to be addressed for the systems with machine-learned components in the feedback loop. To develop a general theory for stability and stabilizability of neural network (NN)-controlled nonlinear systems subject to bounded parametric variations, a Lyapunov-based stability certificate is proposed and is further used to devise a maximal Lipschitz bound for a class of stabilizing NN controllers, and also a corresponding maximal Region of Attraction (RoA) within a user-specified safety set. To compute a robustly stabilizing NN controller that also maximizes the system’s long-run utility, a stability-guaranteed training (SGT) algorithm is proposed. The effectiveness of the proposed framework is validated through an illustrative example.