I think that from a medical perspective your advice was correct, but your physical explanation of why was not.
The wavelength dependence of the extinction due to fog depends on the distribution of particle sizes. If the particles are bigger than the wavelength of light, then the (Mie) scattering and extinction become independent of wavelength, and this I think is the usual case - Price (2010) provide a brief review and some data for fogs and the consensus seems to be that typical droplet sizes are 10-20 microns, though can be bigger or smaller depending on exactly what type of fog/mist it is. Grabner & Kvicera (2011) discuss this in some detail and show (model) curves of extinction versus wavelength that are quite flat across the optical and UV ranges (though do allow more IR light through) unless the fog particle size falls below the wavelength of light.

Now, if you think that the problem with the Sun is due to UV light entering the eye, then the effect of fog would simply be to allow you to expose yourself to more UV light, as the attenuation at other wavelengths would also stop the reflex you would normally have to blink or turn your eyes away or for your pupils to narrow.
Thus although there may not be any acute damage caused by looking at the Sun through fog, you would certainly be increasing your long term exposure to short wavelength radiation if you got into the habit of doing it, possibly leading to cataracts and macular degeneration. The risk is thought to be cumulative. A brief look at some sources suggests that recommended sunglasses block much more (or even all) UVA and UVB radiation (290-400nm) compared to longer wavelength radiation for this very reason. Fog is like "grey" sunglasses, allowing you to damage your eyes without the acute sensations which tell you that damage may be occurring
Absent a contrary medical opinion, I would say that UV light from the Sun is particularly harmful and I would not stare at the Sun through fog.