We show that the proposed measurements of the electric dipole moments of light nuclei in storage rings would put strong constraints on models of flavor-diagonal CP violation. Our analysis is exemplified by a comparison of the Standard Model including the QCD theta term, the minimal left-right symmetric model, a specific version of the so called aligned two-Higgs doublet model, and briefly the minimal supersymmetric extension of the Standard Model. By using effective field theory techniques we demonstrate to what extent measurements of the electric dipole moments of the nucleons, the deuteron, and helion could discriminate between these scenarios. We discuss how measurements of electric dipole moments of other systems relate to the light-nuclear measurements.