Possibly. Stomach acid can have a pH as low as 1.5, nearly as strong as concentrated sulfuric acid. It's usually buffered by the food you eat, that comes up with the vomit. So while in theory, pure stomach acid could burn you, most of the time food and saliva neutralize it somewhat, raising the pH before or while it comes up (and out.) sorry if that was disgusting, but you asked the question.
No. No, stomach acid will not cause spots on the chest.