Previous inverse optimal adaptive controllers (IOACs) have been developed that can handle structured (i.e., linear in the parameters (LP)) uncertainty for a particular class of nonlinear systems. A full-state feedback IOAC is developed in the companion Part I paper for Euler-Lagrange systems with an uncertain time varying inertia matrix. In this paper, an output feedback IOAC is developed to asymptotically minimize a meaningful performance index while the generalized coordinates of a nonlinear Euler-Lagrange system asymptotically track a desired time-varying trajectory despite LP uncertainty. A Lyapunov analysis is provided to examine the stability of the developed output feedback optimal controller, and preliminary experimental results illustrate the performance of the controller.