How to fix water damage to wooden table
Definitely looks like you pulled a surface finish off the wood. Either the original stain was a surface-stain, or the water and cleaning may have pulled the color out of a shallow penetrating stain.

You're best option for a quality finish is going to be stripping back the whole table and starting fresh.

Assuming that's not an option, you could try finding a similar color finish at the hardware/home store. If you know what kind of finish was used originally, you should match it. Wood finishing is as much an Art as a Science, so the usual warning apply, but in this case, an imperfect patch may be better than the bare wood.

Sand away the affected area and an inch or so around it, so you can feather in the patch. This kind of patch will never look perfect, but may be "good enough". Apply your stain with a brush or a clean rag, following the instructions on the container. The longer you leave the stain on the surface, the darker it will be. Once you feel it's matching, wipe away the excess stain. Stain is for color only, and won't provide any long term protection. You can apply stain in multiple passes.

After patching you can blend the patch a little by using one of the antique furniture restorers (Restor-a-Finish), which will also help the look of the rest of the table. Just make sure to wait several days, or you'll pull the new patch out of the wood.

As a note. You can try an all-in-one stain+finish, but these products don't always work, and controlling the color can be difficult. In this case, you may want to skip the actual protective finish coat, and just try to stain it to match. 

As always, try out any finishing on a less noticeable area first, like the inside of a face frame or drawer.