Bird Droppings Ate Through My Clear Coat?!?

I have a silver car and an awful bird blasted my car with cr*p before a snow storm, so the bird droppings sat on my car for a few weeks with snow and road salt piled on top of it. Now that I've washed my car, it looks like it ate through my clear coat. There's a noticeable stain and it looks like light gray.

Is there any way I can fix this on my own, or will I have to take it to a body shop?
 
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