Reynolds-averaged Navier-Stokes (RANS) equations for steady-state assessment of incompressible turbulent flows remain the workhorse for practical computational fluid dynamics (CFD) applications, and improvements in speed or accuracy have the potential to affect a diverse range of sectors. We introduce a machine learning framework for the acceleration of RANS by training an artificial neural network to predict steady-state turbulent eddy viscosities. This surrogate model for the turbulent eddy viscosity is assessed for parametric interpolation, while solving for the pressure and velocity equations to steady state, thus representing a framework that is hybridized with machine learning. We achieve accurate steady-state results with significant reduction in solution time when compared to those obtained by the Spalart-Allmaras one-equation model. Most notably the proposed methodology allows for considerably larger relaxation factors for the steady-state velocity and pressure solvers. Our assessments are made for a backward-facing step with considerable mesh anisotropy and separation to represent a practical CFD application. For two test experiments with different inlet velocity conditions we see a factor of 6.8 and 7.5 decrease in the overall time to convergence. The proposed framework represents an excellent opportunity for rapid exploration of large parameter spaces which prove prohibitive when utilizing turbulence closure models with extra coupled partial differential equations.