This paper is devoted to the maximum pseudo-likelihood estimator of a vector $\Vect{\theta}$ parametrizing a stationary marked Gibbs point process which is not necessarily a locally stable exponential family model. Sufficient conditions, expressed in terms of the local energy function, to establish strong consistency and asymptotic normality results of this estimator depending on a single realization are presented. These results constitute an extension of the ones obtained in \cite{Billiot08} where the local energy function was assumed to be parametrically linear and stable. By applying these tools, we finally obtain the main results: consistency for both the Lennard-Jones model and the finite range Lennard-Jones model and asymptotic normality for the finite range Lennard-Jones model.