Try Restore-A-Finish. I've used it on all kinds of vintage furniture with no ill effects. Sears Hardware and Ace are easiest places in my area to get it. I'd strongly recommend trying it without steel wool first (instructions at web site and on can). If you find you need to apply it with steel wool, make sure you're getting the finest grade (Super Fine #0000).
Link below is to RAF.

Here is a link that might be useful: Restor-A-Finish

Toothpaste is a very mild abrasive that I've used to remove water rings left by glasses on wood tables. Just lightly swirl it on the spot with a rag.

Here are some suggestions from a Handy Man website.

Apply a generous layer of mayonnaise onto the mark and leave over night. In the morning, wipe with a thick cotton cloth such as a cloth diaper.

Polish the mark with toothpaste until it disappears. Be sure to use white toothpaste for this, not a gel.

If there's a smoker in the house, rub the mark with cigar or cigarette ashes until it disappears.

Rub a paste of baking soda and water into the stain. Leave overnight and wipe off in the morning with a cotton cloth.

Good luck!!

Try the mayo trick before anything else.
It actually works.

A white mark on your table tells us that there is moisture under the finish .
Good luck with the "snake oil" remedies . Be careful.

refinishing the table would be the correct way to handle this.

The mayo trick absorbs the water and removes the spot, depending on what the finish is.
If every piece of furniture that got a water spot was refinished it would cost a fortune.

Actually, as an emulsion of egg yolks in vegetable oil, mayo will be hydophobic. It will not suck water out of the wood. It merely adds vegetable oil to the spot which temporarily hides the damaged finish. There is a reason why vegetable oils are NOT recommended for use on cutting boards or other wood surfaces. It is because they can go rancid. The mayo may do the same on your table top in time.

The toothpaste idea is a mild abrasive, so it will abrade the damaged finish. It is mild enough that you can stop before you go too far.

John

No, mayo is an emulsion of vinegar and oil held together by the lecithin in the egg yolks.
It will remove the moisture completely, and is not absorbed by the wood if the finish is otherwise intact.
The mayo does not stay on the tabletop, it is removed after a few hours.
If you have not done it, stop claiming you know it will not work.
I have used it on shellac finishes (works very well) and on nitrocellulose finishes (works but takes longer).

I know an unfinished wood bowl can absorb vegetable oil and develop a rancid smell. But this is finished wood we're talking about on this table and it won't be absorbed.

"Your claim about removing moisture has no basis at all, nor have you suggested a mechanism for this."

Not true.
Mayo is an emulsion using a natural 'soap' called lecithin from the egg yolks.
One side is tied to an oil molecule, the other to the vinegar.
While oil is hydrophobic, the vinegar side is hydrophilic.
This is exactly how an actual soap is made.

I use Mineral Oil on my butcher block. Is this supposed to cause problems, or is it specific to vegetable oils?