You may have a contract regardless of whether it was signed.  Take it
to a business transaction attorney for review.  The issue will be
promissory estoppel, which I would have to brush up on before trying
to explain.  But even if you have a contract, it might be possible to
fire you under the contract.  The attorney will figure that out.
Meanwhile, keep looking.  Finding a position will be better than
litigation.

McGyver

I'm not a lawyer, this is for discussion purposes only.

Did the program director represent that he would/had signed the contract?  
Or, if it was implied (You can't work here without a contract, sign here),
a reasonable person could conclude that the contract existed and was valid
even if the employer never signed it, especially if they required the
resident to abide by the requirements of the contract.

If he wants to stay, he'll probably want to consult with a lawyer
conversant in the employment laws of that state.

Dakshin,

  You provide very little information and I'm no longer current on med
education but here's an opinion. First, of course you can sue. You would be
wise to speak with several attorneys to get an evaluation of your chances
and an estimate of the costs. Many attorneys offer a free initial consult.
Assuming you sue to the residency program and are successful you will still
need to find another residency. A suit may seriously damage your chances for
another residency program. Have your spoken with the department chairman and
the med school dean to get an honest appraisal of your chances of enrolling
in another program or even changing your specialty?
  As you know every year a few residents in any specialty will wash out due
to various factors.  In the past the chairs would find something else (lab
work et c.) until the yearly contract ended and then assist in finding
another program or specialty. I'd be surprised if there is not a common
method is existence to address your problem. Sadly, some individuals are not
cut out for a traditional medical career. The pharmaceutical industry,
medical device manufacturers, the FDA, NIH, et c. all will hire MDs.

Good luck,
Dave M.

Look up some basic contract law. By giving the resident the contract he was
extending an offer, by signing it the resident was accepting the offer. As
long as the contract was drawn up by the director I don't think he has to
sign it for it to be enforcable.

I'm not a lawyer.