Wow, that's wierd. I've seen lots of water spots in finishes but never any
like that. The norm is irregular edges and milky, diffuse white, not solid
white like those; they are caused by water being trapped within the finish
and can generally be removed, lots info about methods via Google.
For your spots, it seems to me that the first thing you need to do is
determine where they are vertically. There are three possibilities...
1. on the finish
2. in the finish
3. under the finish; i.e., the finish is gone and you are seeing what
was under it (which isn't wood)
Your finger tips should tell you which of the three it is.
The second thing you need to know is what the finish is. Again; three
possibilities...
1. varnish
2. lacquer
3. shellac
The most likely is lacquer assuming the table was commercially made. If
home made, most likely IMO is varnish followed by shellac.
If the damage is IN the finish and the finish is either lacquer or shellac
it may be able to be removed using the appropriate thinner (alcohol for
shellac, lacquer thinner for lacquer). DAGS for methodology. For varnish,
a varnish amalgamator might work, don't know; again, DAGS.
If the damage is ON the finish about all you can do is remove it physycally
via gentle and careful scraping.
If it is under the finish - finish is gone - about the only realistic thing
would be to sand down and refinish. It would be possible for someone to use
artists colors and rebuild the appearance of the rest of the table then
clear coat over but it would require someone with *MUCH* experience and
ability and would not be cheap; the same thing could be done if the damage
is on or within the finish, same caveats.
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I notice that in a later post you said the table was always treated with
oil. I don't understand that as it would serve absolutely no purpose on a
table with a clear top coat; all it would do is - maybe - build up a bunch
of gunk.