The contributions dependent on ionic specificity (including those related to the differences in polarizabilities of cations and anions) to the surface tension and ion adsorption on the boundary between an aqueous strong electrolyte solution and the vapor-gas phase are taken into account. The role of these contributions in the thermodynamics of vapor condensation on salt particles completely dissolved in droplets that are emerged on these particles from the vapor is studied. Consistent domains of the applicability of the analytical theory suggesting the complete dissolution of a salt particle and the dissociation of substance comprising this particle into ions, as well as the ideal behavior of solution in a droplet and the linearization with respect to excess electric potentials near the droplet surface, are established in the approximation of a quasi-planar interface. Formulas are derived for the threshold values of the chemical potential of vapor molecules upon the barrierless condensation and for critical vapor supersaturations upon the barrier condensation on salt particles. In the explicit form, these formulas express the dependence of these values on the initial size of salt particles, physicochemical parameters of solution in droplets, and the charge of formed ions. Calculations for water condensation on NaCl, Na₂SO₄, and MgCl₂ particles are performed using these formulas.