Yes you can. If you do not get anywhere using this approach the next thing you would have to do is file a law suit against the at fault driver. However, you may have certain coverages on your own insurance that may pay for your damages. Your best option is to contact an attorney to give you free phone advise because there are several questions needed to be asked of you before an attorney can give you the proper advise. For example, do you have collision coverage, where you injured, do you have uninsured motorist coverage, and so on. Contact an attorney A.S.A.P to see what your best options are.
You can "go after him with phone calls and letters," but that will likely waste your time. Your insurance company should be handling this; if you are not going that route, for whatever reason, send a demand letter with a copy of the repair bill by certified mail, return receipt.
When he fails to pay, bring a lawsuit in small claims court. If outside the jurisdictional limits, you may wish to retain counsel.
The foregoing is for general information purposes and does not establish an attorney-client relationship.
If the at-fault driver was not insured at the time of the accident, it is best for you to go through your own insurance policy if you have the coverage. You should contact your insurance company and find out if you are covered.