Do I have a case?

In order to pursue any personal injury claim, the plaintiff must be able to prove four things: 1. A duty owed, 2. a breach of the duty owed, 3. an injury/damages, and 4. that the injury/damages were caused by the breach of the duty owed (i.e. proximate cause). If a plaintiff fails to prove any of these things, the case will fail.
In the medical malpractice setting, all doctors owe a duty to a patient to treat/care for them within the accepted standards of medical care for that particular field. If a doctor fails to diagnose or discover an injury which was readily apparent, there may have been a breach of that duty. For the sake of argument, we will assume that the doctor in your son’s case breached the duty. So you have established elements 1 and 2.
Where your son has a problem is with elements 3 and 4. First, his broken arm is not the injury, since his arm was broken by the ATV accident, and would have still been broken even if earlier diagnosed by the hospital. Arguably, your son may have an injury in the form of additional pain and suffering caused by the delay in diagnosis, but he would have still suffered from pain even if it would have been diagnosed. So whether you can prove element 3 and 4 is questionable. The same is true of damages. Besides an additional trip to the hospital and the monetary value of whatever additional pain and suffering was caused by the delay in diagnosis, it will be extremely difficult to prove that your son suffered any real damage by the delay in diagnosis. It would be different if the delay in diagnosis resulted in a catastrophic injury, such as nerve damage, paralysis, etc., but that doesnt appear to be the case here. So damages (or the lack thereof) is a serious issue with any potential claim in your case.
Even if you could prove injury and damages, the simple fact is the value of the case is most likely outweighed by the expenses of pursuing the case. Most medical malpractice cases cost in the tens of thousands of dollars to investigate and pursue. Considering the limited amount of damages in this case, the expenses and attorney fees of pursuing a claim would easily exceed the amount that your son could recover if he were successful in pursuing it.
Please note that these are only my thoughts on your son’s claim based upon the limited details you provided. If you feel that you have a claim worth pursuing, I would strongly encourage you to speak with an experienced malpractice attorney as soon as possible, as you have as little as one year from the date of the malpractice to file a lawsuit, or the claim could be forever barred.
Best of luck to you and your son. I hope he has a full and speedy recovery from his injuries.