The rotation curves of spiral galaxies exhibit a diversity that has been difficult to understand in the cold dark matter (CDM) paradigm. We show that the self-interacting dark matter (SIDM) model provides excellent fits to the rotation curves of a sample of galaxies with asymptotic velocities in the 25-300 km/s range that exemplify the full range of diversity. We assume only the halo concentration-mass relation predicted by the CDM model and a fixed value of the self-interaction cross section. In dark-matter-dominated galaxies, thermalization due to self-interactions creates large cores and reduces dark matter densities. In contrast, thermalization leads to denser and smaller cores in more luminous galaxies and naturally explains the flatness of rotation curves of the highly luminous galaxies at small radii. Our results demonstrate that the impact of the baryons on the SIDM halo profile and the scatter from the assembly history of halos as encoded in the concentration-mass relation can explain the diverse rotation curves of spiral galaxies.