We explore the morphology of galaxies living in the proximity of cosmic voids, using a sample of voids identified in the Sloan Digital Sky Survey Data Release 7. At all stellar masses, void galaxies exhibit morphologies of a later type than galaxies in a control sample, which represent galaxies in an average density environment. We interpret this trend as a pure environmental effect, independent of the mass bias, due to a slower galaxy build-up in the rarefied regions of voids. We confirm previous findings about a clear segregation in galaxy morphology, with galaxies of a later type being found at smaller void-centric distances with respect to the early-type galaxies. We also show, for the first time, that the radius of the void has an impact on the evolutionary history of the galaxies that live within it or in its surroundings. In fact, an enhanced fraction of late-type galaxies is found in the proximity of voids larger than the median void radius. Likewise, an excess of early-type galaxies is observed within or around voids of a smaller size. A significant difference in galaxy properties in voids of different sizes is observed up to 2 R-void, which we define as the region of influence of voids. The significance of this difference is greater than 3 sigma for all the volume-complete samples considered here. The fraction of star-forming galaxies shows the same behavior as the late-type galaxies, but no significant difference in stellar mass is observed in the proximity of voids of different sizes.