If client does not accept then its a problem. Best thing to do is understand the gap and ask the contractor to do the correction. Then you can do RCA on why there was a gap ?. If its missed from client side, ask for CR.

Since the contractor has legal binding with you, before going to court try to find ways to resolve the issue, before taking it to court or next level

The customer is the only one that counts! In this situation you would first have to look at the RTM and see if something changed that the customer was not aware of. If not than you would look at the SCOPE Statement and see if he signed off. If yes than he would have to put in a change request for what he didn't get and pay more. Either way you have to resolve in favor of the customer/sponsor. The sponsor would be advising as to if there was money to do more work.

When there is disagreement between the contractor/vendor and the client/customer, always try to 'NAL it down.' NAL is my acronym for

N - Negotiation: this is the lowest level of problem resolution in any relationship, business or personal;
A - Arbitration: if the parties cannot agree on the resolution there is an intermediary step before court; you can enter into 'binding arbitration' or mediation. This can still involve a legal process but is not as severe or costly as a lawsuit.
L - Litigation: a full blown lawsuit. This is the worst case scenario and should be avoided when possible.