The exchange of an axion-like particle between atomic electrons and the nucleus may induce electric dipole moments (EDMs) of atoms and molecules. This interaction is described by a parityand time-reversal-invariance-violating potential which depends on the product of a scalar g s and a pseudoscalar g p coupling constant. We consider the interaction with the specific combination of these constants, g s e g p N , which gives significant contributions to the EDMs of diamagnetic atoms. In this paper, we calculate these contributions to the EDMs of 199 Hg, 129 Xe, 211 Rn and 225 Ra for a wide range of axion masses. Comparing these results with recent experimental EDM measurements, we place new constraints on g s e g p N . The most stringent atomic EDM limits come from 199 Hg and improve on existing laboratory limits from other experiments for axion masses exceeding 10 −2 eV.