Grab a cheap can of paste wax from any hardware store. Rub it into the white ring with a soft cloth, let it sit for a minute, then buff it off. Works on old wood tables and dressers. I fixed my grandma's side table that way and you can't even tell where the glass was.
Wait, you can just fix it like that? I always thought those white rings were permanent, like the wood was actually stained. My mom's coffee table has had one for years and we just put a coaster over it.