Before going to the extreme of using sandpaper or fine steel wool to buff it out try these Eco Friendly home remedies.
Sometimes the answer to the remedy is not found at your local hardware store but in your refrigerator.
Spray vegetable cooking spray on the white marks and let it set for a few minutes. Rub the excess off with a paper towel In about 6 hours it should be gone!
Toothpaste is also know to do the trick on water mark stain removal. Apply a small amount of toothpaste (not the gel kind) by using a damp clean cotton cloth. Rub the toothpaste into the water stain and polish the wood. A paste of baking soda and water has worked as well. Gently rub paste into the stain and leave on for about 30 – 40 minutes. Remove the paste with a clean dry cloth then polish with an Eco friendly wood polish
Apply a coat of mayonnaise over the stain and let sit over night. The next morning, wipe off with a clean, dry cloth and polish with an Eco friendly wood polish.
Some people swear over mayonnaise and cigar or cigarette ashes mixture. Rub it into the mark and it should buff off with polishing.
I know these remedies sound outright crazy but it is better than taking a sander out and sanding the whole finish off your table.