Will GFCI curcuit breaker operate properly with neutral pigtail wire not connected?
The GFCI absolutely needs its neutral wires

He can't make it work and he's trying to walk out the door.  He needs your signature on a piece of paper, so he futzes with wires until it works, fumbles an excuse, gets you to sign, gone.   What are you to say to his face? 

In effect, he challenged you to either accept things as-is or be discourteous.  You chose to be courteous.  Standard contractor play.

If you read the instructions for the GFCI breaker, he certainly did not install it according to the instructions, which means the work is illegal under NEC 110.3b ( listed or labeled equipment shall be installed and used in accordance with any instructions included in the listing or labeling ). 

There's a real problem, likely a hard one

Most likely he found to his dismay that the circuit's neutral was being shared by another circuit... either a load grabbed this hot and another circuit's neutral, or vice versa.  Or alternately, one of the loads on that circuit actually does have a ground fault -- people always forget about that possibility, but that is the entire point of a GFCI, after all!  He didn't want to spend half a day doing a "bug hunt", so he dodged the issue and got paid. 

Sometimes it is a systematic version of that thing, called a multi-wire branch circuit -- but those are easy for any competent person to spot.  And very easily fixed: fit a 2-pole GFCI and done.  So I doubt it's that.