The good news is that a white spot indicates that the damage (water) is in the finish, not in the wood.
To try to remove them, do the following, in the following order:
Milky-white watermarks, arranged in order from the least damaging (and generally least effective) to the potentially most damaging.
• Apply an oily substance, such as furniture polish, petroleum jelly or mayonnaise, to the damaged area and allow the liquid or gel to remain overnight. The oil will often restore some of the transparency (by filling some of the microscopic voids) but seldom all of it.
• Heat the finish with a blow dryer or heat gun to soften the finish so it consolidates.
This may restore some of the transparency if you get the temperature just right , but if you get the finish too hot, it will blister. Avoid getting the finish any hotter than is comfortable to touch.
• Dampen a cloth with denatured alcohol and wipe gently over the damaged area. Wipe using a motion like a plane coming in for a landing and taking right off. The trick is to dampen the cloth just enough so it leaves the appearance of a comet’s tail of evaporating alcohol trailing as you wipe. (You can practice by wiping across a more resistant surface such as polyurethane or plastic laminate.)
If you get the cloth too wet, the alcohol may soften the finish too much and dull the sheen or smear the finish. This is especially likely if the finish is shellac (used on most furniture finished before the 1930s), but this technique is most effective on shellac.
• Cut through the damage by rubbing with a mild abrasive such as toothpaste, or with salt mixed with olive oil. Fine #0000 steel wool lubricated with a light oil, such as mineral oil, is more effective because it cuts faster, but steel wool will leave noticeable scratches in the surface. Use steel wool only as a last resort.
Rub the damaged area until the water damage is gone, being careful not to rub through the finish. Then, if the sheen is different from the surrounding area, even it by rubbing the entire surface with paste wax.
Bob "Boardman" Borders