Mr. X,
I am not sure what can be the problem here. But, when we install gutters, they usually tuck way up behind the roof edge. There is actually a flange that should attached to the roof under the shingles that runs into the gutter. The best solution that I have is to have the contractor come back and install flashing the runs under the shingles into your gutter. I truly believe that edging should not be used on the bottom of the ridge, just a this flashing, and then the shingles should hang over that about an inch and drip into the gutters.
hope it helps, in any form, you should not pay the agree amount until you are happy with the job. If he is unwilling to work with you now, what would happen if you need him back in a month or two?