Fill a spray bottle with cool water and aim it at the etch marks. Apply a thin and even coating of water to the marks.
Sprinkle a small amount of polishing powder on the marble. Rub a damp cloth over the polishing powder, rubbing off the etched marks. Alternatively, spray more water on top of the powder and buff with a dry cloth.
Check the marble for any remaining alcohol marks. Attach a buffing pad to a power drill if the stains still appear.
Apply more water to the stains and sprinkle the powder over the top. Turn the drill to its lowest setting and aim the buffing pad at the mark.
Hold the buffing pad against the powder and gently buff out any alcohol marks. You may need more powder and water before the mark completely disappears. Wipe down the marble with a damp cloth, removing any marble dust.