Even though this idea sounds nice and I do enjoy eliminating work, I don't think the refrigerator is compatible with a set of spray arms and hot water.

I think once you've taken most of the stuff out of the fridge, you've done most of the work. I guess you could wash with the ketchup in, but it would get hot.

Besides, the dishwasher wastes a large amount of space for the spray arms, but they really don't keep the inside of the dishwasher all that clean. And I think that the hot water has more effect cleaning the gunk off of plates than the spray-it would be hard to direct the sprays in a refrigerator to get all of the shelves, the butter-door, the meat drawer, etc. You'll have to redesign to eliminate these features, but I like them.

So I guess I'm against it, though I can see why you thought it up.

A dishwasher is designed to wash the dishes, not the dishwasher itself. If the system was designed to wash the fridge rather than its contents, I think it could be made pretty effective.

Don't count this idea out so quickly. It could be possible with sonic cleaning, or acoustic cleaning(wiki). Supposedly as long as you're cleaning dry materials this would work. And I think that ice is technically dry so it might even work as deicing.