If you really did just get the bill it will be several months (and several more copies of the bill sent to you) before a final bill would show up that would indicate the account is "PAST DUE" and heading for collections.
IANAL but you can safely wait for the itemized bill.
In the future to save on costs, you might try to find a 24-hour Urgent Care instead of full-fledged ER. Unfortunately the level of equipment that they have to keep in a full ER means just having someone in a bed costs them a lot of money, even if they are just swabbing your throat for a strep test.