From reading the article it sounds to me like this only effects public NY emergency rooms? So if you have terminal cancer or break your leg you would most likely be admitted to an oncology service or orthopedic service and be able to receive appropriate pain management for your condition. I also assume when the article states patients will no longer be able to get more than three days of narcotic pain relievers that means on discharge from the emergency room. Most indications that necessitate chronic therapy with long-acting narcotics would require an admission to the hospital or follow up in an outpatient clinic. So you leave the emergency room with a few pain pills until you can seek more qualified/appropriate care for whatever is wrong with you.
Another thing to think about is hospital accreditation. The Joint Commission has standards on all sorts of things hospitals must do to be accredited and subsequently receive Medicare/Medicaid dollars. One of those standards is pain. Most of the hospitals that this will effect probably rely heavily on Medicare/Medicaid money, so they have an incentive to alleviate a patient's pain.
I'm not saying I think Bloomberg (I can't stand the guy) is right. I'm just saying it may be an appropriate step to help decrease prescription drug abuse.