A (Formula presented.)-continuous rational cubic fractal interpolation function was introduced and its monotonicity aspect was investigated in [Adv. Difference Eq. (30) 2014]. Using this univariate interpolant and a blending technique, in this article, we develop a monotonic rational fractal interpolation surface (FIS) for given monotonic surface data arranged on the rectangular grid. The analytical properties like convergence and stability of the rational cubic FIS are studied. Under some suitable hypotheses on the original function, the convergence of the rational cubic FIS is studied by calculating an upper bound for the uniform error of the surface interpolation. The stability results are studied when there is a small perturbation in the corresponding scaling factors. We also provide numerical examples to corroborate our theoretical results. © Springer India 2015.