The present article is concerned with the Lyapunov stability of stationary solutions to the Allen-Cahn equation with a strong irreversibility constraint, which was first intensively studied in [2] and can be reduced to an evolutionary variational inequality of obstacle type. As a feature of the obstacle problem, the set of stationary solutions always includes accumulation points, and hence, it is rather delicate to determine the stability of such non-isolated equilibria. Furthermore, the strongly irreversible Allen-Cahn equation can also be regarded as a (generalized) gradient flow; however, standard techniques for gradient flows such as linearization and Łojasiewicz-Simon gradient inequalities are not available for determining the stability of stationary solutions to the strongly irreversible Allen-Cahn equation due to the non-smooth nature of the obstacle problem.