After a car accident (that wasn't my fault) I went to a personal injury lawyer. He said he would refer me to one of his doctors. Should I use his doctor?

No.  If you have a serious injury caused by a car accident, or any other type of accident for that matter, there is no way you should be going to one of "his" doctors.  How do you know this is the best doctor to treat your injuries - is he or she a specialist with a focus on serious back injuries, such as an orthopaedic surgeon or neurosurgeon with a focus on treating the spine, or a foot and ankle specialist, upper extremetiy, hand surgeon - you get the picture.  The point is you need to research and find the best doctor for your specific injury or condition.  What if you're not happy with the doctor your lawyer sends you to?  How will the doctor get paid - will he wait until your case settles and then demand the full amount of the bill?  

As you can see, there are a lot of problems with this idea.   Do your own independent search for a lawyer and a doctor if you're hurt in an accident.  You'll be better off for it.