I have a friend who uses apple cider vinegar for her GERD. Works great but tastes really bad
Dogs don't seem to mind it, though. However, since it's so dilute in the water, I'd really not expect it to have much effect on the urine. So if it works, you for sure have to let us know, klm!
Dog urine has high pH, which makes it a base. It also contains nitrogen, which is a plant nutrient. So if you can bring the pH down to 7 (neutral), which is what I'm thinking the vinegar (an acid) is supposed to do, and eliminate other sources of nitrogen to Gryphon's 'pee place'
(for instance, if you fertilize the rest of the lawn,
don't fertilize that spot), you might be able to coax some sort of ground cover there, especially if you can find a way to keep it well watered without creating mud.
If you find something that works, let me know. In winter, the Packmembers all have their own favorite areas to pee in and it always takes us still about July to get the grass cover back to where it should be.