We apply the modified acceleration law obtained from Einstein gravity coupled to a massive skew-symmetric field F_{mu}{nu}{lambda}_ to the problem of explaining galaxy rotation curves without exotic dark matter. Our sample of galaxies includes low surface brightness (LSB) and high surface brightness (HSB) galaxies and an elliptical galaxy. In those cases for which photometric data are available, a best fit via the single parameter (M/L)_stars_ to the luminosity of the gaseous (HI plus He) and luminous stellar disks is obtained. In addition, a best fit to the rotation curves of galaxies is obtained in terms of a parametric mass distribution (independent of luminosity observations) - a two-parameter fit to the total galactic mass (or mass-to-light ratio M/L) and a core radius associated with a model of the galaxy cores - using a nonlinear least-squares fitting routine including estimated errors. The fits are compared to those obtained using Milgrom's phenomenological MOND model and to the predictions of the Newtonian/Kepler acceleration law.