This article is concerned with the rigorous justification of the hydrostatic limit for continuously stratified incompressible fluids under the influence of gravity. The main peculiarity of this work with respect to previous studies is that no (regularizing) viscosity contribution is added to the fluid-dynamics equations and only diffusivity effects are included. Motivated by applications to oceanography, the diffusivity effects included in this work are induced by an advection term whose specific form was proposed by Gent and McWilliams in the 90s to model effective eddy correlations for non-eddy-resolving systems. The results of this paper heavily rely on the assumption of stable stratification. We provide the well-posedness of the hydrostatic equations and of the original (non-hydrostatic) equations for stably stratified fluids, as well as their convergence in the limit of vanishing shallow-water parameter. The results are established in high but finite Sobolev regularity and keep track of the various parameters at stake. A key ingredient of our analysis is the reformulation of the systems by means of isopycnal coordinates, which allows to provide careful energy estimates that are far from being evident in the original coordinate system.