The problems of stability and stabilization are addressed for a class of nonlinear mechanical systems with distributed delays. Assuming that potential and kinetic energy functions are homogeneous of different degrees, it is shown that the global asymptotic stability of the zero solution for an auxiliary delay-free nonlinear system implies the local asymptotic stability for the original model with distributed delay. The influence of additional nonlinear and time-varying perturbations is investigated using the averaging techniques. The results are obtained applying the Lyapunov–Krasovskii approach, and next extended via the Lyapunov–Razumikhin method to the case with negligible dissipation. The efficiency of the proposed theory is illustrated by solving the problem of a rigid body stabilization.