You can correct chromatic aberration computationally by realigning red/green/blue layers. However, like correcting geometric distortion, those corrections usually are not by whole multiples of pixels and thus have to distribute the light on one source pixel to at least two target pixels. This causes a loss of sharpness. If you try countering this by resharpening afterwards, you amplify noise and are prone to halos.
So far this does not sound worse what distortion correction already does and you can basically combine the corrective actions of distortion correction and chromatic aberration correction before resampling to a rectangular grid in order to get less cumulative blurring than if you resampled independently.
So far, so bad.
The next problem is that chromatic aberration comes in two flavors. What I talked about just now only deals with lateral chromatic aberration which tends to be stronger the more you move from the center. There is also longitudinal chromatic aberration with its main consequence being purple fringing: if you photograph a tree's branches against a backdrop of a blue or clouded sky, significant amounts of ultraviolet and near-ultraviolet light are detected by the blue sensors. Longitudinal chromatic aberration means that this light typically is bent stronger than other light, putting its focusing plane before the sensor. This leads to unsharp purple halos around branches to both sides assuming that the branches are in-focus. If they are out-of-focus, the bluish components may actually be in-focus, giving slight red fringing (you rarely see that since it requires the focus to be too short in the first place). How much of those purplish unsharpness appears depends on the distribution of wavelengths hitting the blue sensor. Indoors LED and fluorescent lights will be harmless in comparison, incandescent light usually at least is colder (regarding color temperature rather than painter terminology) than sunlight.
Which brings us back to lateral aberration: it is not just the blue sensor which is receptive to several different wavelengths: all sensors detect a whole range of wavelengths with different sensitivities, and chromatic aberration hits all of those wavelengths differently, causing the signal of each sensor to be not just moved but also spread out according to the distribution of wavelengths hitting it. What distribution would that be? Different white balance settings take a guess at wavelength distributions but that guess is focused on getting the balance between three primary colors right.
Guessing the right amount of unsharpening means getting the balance of more than just three primary colors right, and that balance changes a lot more across the scene than basic white balance would.
So while you can statistically more or less guarantee that your fringing mostly averages out to average to gray in all directions, a sharp black-and-white edge, when looked at closely, will still kind of resolve into a small rainbow due to different (and only statistically predictable) amounts of unsharpness.
Lens corrections of chromatic aberrations don't work just with a plane of info but are three-dimensional constructions that can be calculated to bring a continuum of wavelengths mostly to the same spot on the same focusing plane. This kind of correction is not possible to do with the 3-band reduced data from a single focusing plane because it just does not contain the same amount of information.