We propose a convergent implicit stabilized finite element discretization of the nonstationary incompressible magnetohydrodynamics equations with variable density, viscosity, and electric conductivity. The discretization satisfies a discrete energy law, and a discrete maximum principle for the positive density, and iterates converge to weak solutions of the limiting problem for vanishing discretization parameters. A simple fixed point scheme, together with an appropriate stopping criterion is proposed, which decouples the computation of density, velocity, and magnetic field, and inherits the above properties, provided a mild mesh constraint holds. Computational studies are provided. © 2010 American Mathematical Society.