This paper gives a theoretical background for a new electrowetting system based on interfaces between two immiscible electrolytic solutions. It presents a linear-response Poisson-Boltzmann theory to describe an electrolytic droplet on a charged flat electrode, bounded by another electrolytic solution. Immiscibility of the two solutions causes back-to-back double layers to form at the liquid-liquid interface, which dramatically change the polarization response. Useful approximations are developed that apply to droplets with typical experimental volumes. Under the derived approximations, minimization of the free-energy functional proves that polarized droplets take the shape of truncated spheres and reveals a law of contact-angle variation with applied potential. This dependence is determined by the interfacial tensions and the electrolyte concentrations in and dielectric constants of the liquid phases. The study of contact-angle variation with electrode potential may be used, among other applications, as a new tool to investigate the effect of solution properties on liquid-liquid surface tensions.