At the risk of being pedantic, dispersion and refractive index are related--generally, the higher a material's RI, the more likely it is to have greater dispersion as well. But dispersion is indeed a distinct phenomenon, best described as variable RI as a function of wavelength.
What makes correction of CA possible is the combination of two materials with different dispersive qualities in such a way that there is still a net refractive power. This was traditionally accomplished using a crown/flint glass doublet and is called an "achromat" design. The correction occurs near the extremes of the visible spectrum, so that red and blue wavelengths converge at the image plane. But green wavelengths are not made to converge at the same point with this design. This explains why we see the green/magenta "bokeh fringing" in out of focus areas of wide-aperture shots. The achromat design puts red+blue = magenta, versus green on opposite sides of the focal plane.
This is where a material like fluorite comes into play. Fluorite exhibits what is called extraordinary partial dispersion. It is a low-dispersion material, but more than this, its dispersion as a function of visible wavelength is different than for typical glasses. If you plot the dispersion as a function of wavelength, the curve is much flatter whereas the curve for a typical glass is more sloped. As a result of the different shape of these curves, it is possible to construct an optical group that corrects the secondary spectrum to a greater extent than possible with the flint/crown doublet. In essence, one calculates the necessary curvatures to focus three wavelengths together. This is called an "apochromat" design.
With increasing optical complexity and variety of materials, it is possible to construct a "superachromat" design that corrects secondary spectrum at four points in the spectrum. But these are extremely costly and the benefit in most photographic applications is small in comparison. One application of such highly correct lenses, however, is in infrared photography, where correction for the visible spectrum is often done at the expense of the IR spectrum, making many lenses unsuitable for IR imaging. Greater correction in this portion of the spectrum would make a lens suitable for both visible and IR applications.
Although Canon is the only manufacturer to use crystalline fluorite elements to control CA, I presume other optics manufacturers have used fluorite-doped glasses to great effect. Sadly many of Canon's lenses, although employing fluorite, don't seem to correct very well for longitudinal chromatic aberration, and I believe this is in part due to other design considerations. A true "APO" lens should display almost no noticeable transverse CA and very minimal longitudinal CA (green/magenta bokeh fringes in high-contrast, out of focus areas).