No, this is not their call, as it involves the safety of everyone on board, and there are federal laws on this. If a passenger is unable or unwilling to perform the duties required in an emergency, they can not sit an exit row, period. If the person clearly couldn't do this, and it was obvious, both the GA and FA were not doing their jobs, and both should be written up. This is definitely a case where you should be writing in - if there was an incident where the exit doors were needed, and this person was not able to open them, not only would this be a problem, but they'd likely be impeding others from being able to do so.

This is a case where letting the FA know immediately (and discreetly), most certainly before the door closed, is a necessity. That nothing happened during the flight is irrelevant.