I don't know why it would or should. Calcium carbonate is a salt composed of the carbonate anion with calcium, which has a limited ability to dissolve into water. The carbonate anion in water will grab hydrogen ions, becoming carbonic acid, and that will dissociate into water and carbon dioxide gas. Fish don't breathe carbon dioxide. This process will also raise the water's pH, since it's taking hydrogen ions (acid) out of the water, and of course it also releases calcium ions into the water. I suppose that fish might be better at taking in oxygen in water that is a little less acid, or which has calcium in it.

I can imagine that calcium and carbonate that are in the water already could bind to the calcium carbonate and precipitate out. This would drive the reaction in the opposite direction, removing carbon dioxide from the water. You can see why this would be good, as carbon dioxide is a waste product, and removing it adds more room for oxygen. However, normally calcium carbonate would be dissolving, not accumulating, except in very warm water. This is why coral reefs only grow in the tropics, where the water is warm enough to precipitate calcium carbonate. But I doubt that would be possible in your fishbowl; goldfish aren't exactly tropical fish.