Is a hospital at fault for not performing appropriate tests regarding an emergency injury?

There are several issues: 
1) First, whether the ER not doing "diagnostic testing" was actually careless or negligent. It may have been; or it may not have been. That depends on what the average reasonable, well-trained, not-careless doctor would have done if presented with a patient with your symptoms. If most doctors would have given you pain medication and sent you home, they did nothing wrong; only if they were careless--that is, they did not do what a reasonable doctor would do--might they be liable. Doctors or an ER don't have to be perfect; they just have to do what is reasonably expected of them.
2) How were you hurt by their failure to diagnose you that first time? IF what they did was careless or negligent, then they could be liable for additional harm or medical costs you incurred due to the delay. The would not be liable for harm or costs you would have incurred no matter what, since those were not caused by them.
3) In considering the harm you may have suffered from the delay in diagnosing you, you may NOT consider any delays caused by you not having medical insurance. That is not the ER's fault or issue: they have no control over your insurance, your finances, whether you choose to go for medical care despite having no insurance or choose to suffer because you don't have it, etc. You can only consider the delay in diagnosis that would have occured between when you went to the ER and when the condition might have been found if you had gone in for follow-up when it became clear the problem was not resolving. Your financial issues or insurance issues are your issue, not the ER; you cannot hold them liable for it.
The above three concerns are what will determine if you have a case and what it might be worth.