I afraid to say the sulfuric acid in the battery has eaten though the lacquer on your oak dining room table.
The only way the fix it properly is the strip the entire table back to the bare wood and re-lacquer your table.
For now as a temporary fix, is to keep putting timber oil on the damaged area so it’s not so noticeable.
Don’t forget to have a look at your house hold contents insurance to see if you have accidental damage cover. You might be able to get them to pay for your repairs.
If you would like to fix your table yourself the site below gives you good instructions.
http://www.oldhouseweb.com/stories/Detailed/10282.shtml
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