A model which incorporates the effects of electron cyclotron current drive (ECCD) into the magnetohydrodynamic equations is implemented in the NIMROD code [C. R. Sovinec et al., J. Comput. Phys. 195, 355 (2004)] and used to investigate the effect of ECCD injection on the stability, growth, and dynamical behavior of magnetic islands associated with resistive tearing modes. In addition to qualitatively and quantitatively agreeing with numerical results obtained from the inclusion of localized ECCD deposition in static equilibrium solvers [A. Pletzer and F. W. Perkins, Phys. Plasmas 6, 1589 (1999)], predictions from the model further elaborate the role which rational surface motion plays in these results. The complete suppression of the (2,1) resistive tearing mode by ECCD is demonstrated and the relevant stabilization mechanism is determined. Consequences of the shifting of the mode rational surface in response to the injected current are explored, and the characteristic short-time responses of resistive tearing modes to spatial ECCD alignments which are stabilizing are also noted. We discuss the relevance of this work to the development of more comprehensive predictive models for ECCD-based mitigation and control of neoclassical tearing modes.