We extend our previous study of surface tension of ionic solutions and apply it to acids (and salts) with strong ion surface interactions, as described by a single adhesivity parameter for the ionic species interacting with the interface. We derive the appropriate nonlinear boundary condition with an effective surface charge due to the adsorption of ions from the bulk onto the interface. The calculation is done using the loop-expansion technique, where the zero loop (mean field) corresponds of the full nonlinear Poisson-Boltzmann equation. The surface tension is obtained analytically to one-loop order, where the mean-field contribution is a modification of the Poisson-Boltzmann surface tension and the one-loop contribution gives a generalization of the Onsager-Samaras result. Adhesivity significantly affects both contributions to the surface tension, as can be seen from the dependence of surface tension on salt concentration for strongly absorbing ions. Comparison with available experimental data on a wide range of different acids and salts allows the fitting of the adhesivity parameter. In addition, it identifies the regime(s) where the hypotheses on which the theory is based are outside their range of validity.