This paper studies the stability problem for a class of nonlinear systems with distributed delay. Using special constructions of Lyapunov–Krasovskii functionals and the differential inequalities method, sufficient conditions are obtained ensuring that zero solutions of the investigated systems are stable with respect to all variables and asymptotically stable with respect to a part of variables. This result is an extension of well-known Lyapunov–Malkin Theorem corresponding to the critical (in the Lyapunov sense) case where the matrix of the associated linear approximation system admits several zero eigenvalues. In addition, some scenarios are considered for which the derived stability conditions can be relaxed. Two examples of applications of the developed theory to the stability analysis and control synthesis for mechanical systems are provided.