This paper develops a systematic method for global asymptotic stabilization (GAS) of general nonlinear control systems with stable free dynamics. The approach is based on the development of non-affine passive systems theory, the idea of feedback equivalence and the technique of bounded state feedback. Using the approach presented here, we show that, as in the case of affine systems, a nonlinear system whose free dynamics are stable is GAS if it satisfies appropriate controllability-like rank conditions characterized by the Lie brackets of affine-part vector fields. We then show how a globally stabilizing smooth state feedback control law can be constructed explicitly. As an application, a dynamic state (respectively output) feedback compensator is presented for a class of input-output non-affine systems.