What rights do I have against an ER doctor for a misdiagnosis?

If the doctor committed malpractice, he--and possibly the hospital--could be liable (or responsible) for any additional medical costs you incurred due to the failure to diagnose the injury correctly the first time, and also for any lost wages and possibly any pain and suffering, again which flowed from or were caused by the misdiagnosis.

So even before considering whether this may have been malpractice (see below) the first issue is whether, if it was, it caused you any injury or cost. If you would have ended up paying the same or nearly  the same for medical care if the diagnosis had been made correctly the first time, would not have suffered any meaningful delays in healing or additional injury or disability, would not have missed additional work, etc., then it's not worth suing. That's because the U.S. legal system only provides compensation for actual losses or injury, so if the malpractice did not cause any additional injury or cost, you can't effectively recover any compensation. (And also: malpractice cases are among the most expensive ones to bring, since you need to pay for a medical expert to provide testimony and evidence.)

If there were injuries or costs caused by the misdiagnosis, then you may be able to recover them if the misdiagnosis was due to malpractice. Malpractice, or "bad practice," is medical care which does not meet then-acceptable standards. Basically, if the ordinary or average doctor should either have notice the severed tendor or at least run additional tests (e.g. X-rays) when you went to the emergency room, but his ER doctor did not, that may be malpractice. On the other hand, if the ER doctor did everything expected of him and came to a reasonable conclusion based on the symptoms and evidence, it's probably not malpractice. Doctors are not required to always be right--just to provide acceptable or reasonable care.