It doesn't, unless you are wealthy, otherwise you try and 'walk off' any issues until they become obviously life threatening, then you go to the hospital and if you have insurance, you hope the insurance will approve the operation you need, and if you don't have insurance you hope the cost wont bankrupt you.
https://en.wikipedia.org/wiki/Emergency_Medical_Treatment_an...
As for other doctor's offices or planned procedures, I think they already can decide to not work with you if you don't pay them. Maybe there's something similar for general practice stuff related to Medicare/Medicaid as well but for exclusively private pay practices I can't imagine something forcing those providers to offer non-emergency services.
However, it doesn't specify what treatment will be provided, so more likely, hospitals would use that sort of information to limit treatment to ER-level interventions and then recommend a wait-n-see approach as followup.