How do I cover up the white spots on my cast iron stove?

Either something has accumulated on the stove or something was taken off the stove by the acids and sugars in the peaches. If you can't clean the spots, then you're looking at the latter situation.

Your stove has some type of finish on it. As Fisher stoves seem to be out of production, official information seems to be sparse. You could apply some type of high-temperature paint, of which there are many different color depths and finishes. Most products I found in 5 minutes of googling seemed to recommended doing the whole stove at once. This could be good for sprucing up the whole stove or could be more work, time, waiting and fumes than you're interested in.

You could also go for a polish or blackening agent, which works more like seasoning a cast iron pan – an organic, lipid wax or oil is applied gradually in layers and the heating and cooling of the stove transforms it into a polymer, which gives it that dully shiny hardened finish. If I had to guess, I would guess that an old school stove is more likely to have a seasoning based finish than a pain.

There are a lot of websites dedicated to wood burning stoves, such as Hearth.com and woodheat.org. With their help and a little research, you could probably identify the exact model of Fisher you have and the year or era it was made; or maybe someone can find a manufacturer recommended product directly.