A Dirac electron system in solids mimics relativistic quantum physics that is compatible with Maxwell's equations, with which we anticipate unified electromagnetic responses. We find a large orbital diamagnetism only along the interplane direction and a nearly temperature-independent electrical conductivity of the order of e^{2}/h per plane for the new 2D Dirac organic conductor, α-(BETS)_{2}I_{3}, where BETS is bis(ethylenedithio)tetraselenafulvalene. Unlike conventional electrons in solids whose nonrelativistic effects bifurcate electric and magnetic responses, the observed orbital diamagnetism scales with the electrical conductivity in a wide temperature range. This demonstrates that an electromagnetic duality that is valid only within the relativistic framework is revived in solids.