Charge-to-orbital conversion via the orbital Rashba-Edelstein effect represents a key functionality for orbitronics but has been challenging to identify. Here, we combine first-principles density functional theory, linear-response theory, and magneto-optical modeling to reveal how this effect can be detected optically through the quadratic magneto-optical Voigt effect in magnetic/nonmagnetic heavy-metal bilayers. We find that, in a cobalt/platinum bilayer, the current-induced orbital angular momentum can exceed the spin contribution by nearly a factor of three and produce a strong optical signal in addition to the equilibrium Voigt effect. Our atom-resolved study reveals that the platinum layer, despite being nominally nonmagnetic, can contribute strongly because of proximity-induced and current-induced magnetic moments. These results establish magneto-optical detection as a route to probe interfacial orbital phenomena in magnetic heterostructures.