Select a vinegar-free polishing solution that is specific for marble floors. Certain polishing solutions are effective for removing etching from marble. The solution must be vinegar-free because the acidic content in vinegar -- and the cleaning agents that contain vinegar -- can eat away at the marble, causing the stone to fracture and flake away.
Mix the polishing solution into a bucket of water. The directions on the solution will tell you how much water to mix, but generally you add one part solution to two parts water for cleaning agents.
Dip a soft handheld sponge into the bucket. Squeeze the excess liquid out over the bucket and apply the sponge to the spot with the etching. Rub the sponge over the etched spot in a back and forth or circular motion. Keep the water and solution contained to just the area over the etched spot.
Use a soft sponge mop for polishing large surface areas of marble. You don't want to use a harsh mop or else this could potentially scratch the surface of the marble as you rub the mop back and forth. Repeat the steps for the mop as you would with the sponge, by dipping the mop into the solution, squeezing out the excess and mopping the etching repeatedly.
Wet one of the clean rags after you are done polishing the etching out of the marble. Wipe the area with the wet rag to remove any residue from the polisher. Use the second clean rag to dry the area. Don't leave the marble surface wet or else this can cause more etching.