Nothing bad will happen to yourself if you leave pink eye untreated however if you don't treat the pink eye with antibiotics you could spread the pink eye onto other people so they would likely get the Pink Eye from you.
If you apply antibiotic ointment to your eye then the Pink eye will stop being contagious within 24 hours and it will usually go away much sooner.
However you don't have to treat pink eye if you don't want too and the pink eye will usually go away on it's own within 3 to 4 days without medical treatment.