“Can you leave the ER without being discharged?”

Yes, you can. The doctor will advise you of the risks of leaving against medical advice. You need to sign a release acknowledging that you have been advised to stay but elect to leave against the advice of the treating physician.

Sure, you can leave the hospital any time. It would be considered “eloped” or “AMA” and possibly insurance may not pay for visit and send you a bill. Probably not a great idea.

NO, BUT CAN LEAVE AGAINST MEDICAL ADVICE SIGNED BY THE PATIENT AND WITNESSED.

You can leave the ER without being discharged; however, it's foolish to leave without finding out what is really wrong with you, or what is causing your pain, or before receiving your medication prescriptions. Before leaving the ER, ask that the physician (or your ER nurse) to explain to you (in terms you understand) what is wrong, what is causing your pain, and talk to you about what medications are being prescribed for you.

Yes AMA, and admitted

It is not the military or prison,  as long as you are competent and not contemplating self- (or other person) injury, you are free to do what you want.  You usually cannot take an inserted IV line with you,  though. The "dwell time" in the ER depends upon many things,  NONE of which do the ER doctors or nurses have ANY control over,  and yet often receive the frustrated patient's scathing diatribe over. There is no ER-mind-control over other departments (lab, imaging, consultants, pharmacy--who each, in their work space have competing demands to prioritize and process), the "next patient" that can always appear at any time, very self-centeredly attempting to die and thus demanding the complete attention of several ER team members for an unknowable duration of time until "stable" in either sense of the word, and we can pick back up (ideally) where we left off with everyone else.  The paperwork components,  so your insurance does not arbitrarily deny your visit and leave you with the large (undiscounted) bill, are onerous and require some focused time along the way, as there is never a predictable down time to finish it after you are discharged, but if incomplete not done may cost you.  Prescriptions and follow up / cautionary guidance take time to generate. Etc., etc.

Keep in mind, that "tv time" is not reality based, otherwise that CPR the Paramedics brought in,  and took us from your care, would be revived, diagnosed with and cured of whatever failed him/her at the time, and be ready for discharge ahead of you,  since they got our collective focused attention (as above). Also, if you have a Primary Care Provider,  the time frame for comparison is different: couple of days to weeks to get in, orders for outpatient testing,  some of which needs scheduled,  then return visit to review and see if you are better, worse, or the same. This leads to the next round of testing,  and/or consulting,  and on and on. Each step can be a day to weeks long. The ER can often jumpstart the process several steps, but will likely take most of the day to accomplish.  And, this assumes that you have come in early in the day on a weekday that is not a holiday, so testing and specialists are readily available.  Everything is not the same evening or overnight hours,  weekends or holidays.  Bring a book or something else to pass the time with during your stay. Also, EAT if coming near a mealtime, the staff often does not get a break to eat, and it is admittedly NOT the emphasis of a location with "Emergency" in the name. Also, there is likely "no stars" rated for our "box lunch" cuisine.

If you keep your expectations realistic,  and are a "patient patient," your visit with the dedicated staff of the ER will be more palatable, and a few more minutes to get the discharge paperwork will not seem like such a long time.