We predict that current perpendicular to the plane giant magnetoresistance (GMR) in a phase-coherent magnetic multilayer is suppressed when one of the contacts is superconducting. This is a consequence of a superconductivity-induced magnetoresistive effect, whereby the conductance of the ferromagnetically aligned state is drastically reduced by superconductivity. To demonstrate this effect, we compute the GMR ratio of clean (Cu/Co)(n)Cu and (Cu/Co)(n)Pb multilayers, described by an ab initio spd tight-binding Hamiltonian. By analyzing a simpler model with two orbitals per site, we also show that the suppression survives in the presence of elastic scattering by impurities.