Response to referee after rejection I recently had a paper rejected: both referees pointed to a unique issue with our work. One referee's criticism was fair, hence I feel the rejection decision was fair. 

The other referee however, seemed to appreciate the paper and know the area extremely well (and thus, I expect to have them as a referee for any future submission). Their entire objection was based on an incorrect assumption -- if their assumption were true, I too would have suggested rejection. However, our results clearly showed this assumption is not true. Thus, their entire basis for rejection could be 1) very easily refuted in a response letter/revision and 2) may lead to future rejections if the referee is not corrected. I am wondering: 

Would it be appropriate to write a "response to reviewers" document to one referee to point out this issue? Is this at all common? I would not be appealing the rejection per se, but rather hoping that the referee better understood our work (with respectful tone and appreciation for the thought they put into their review). If not, should I simply move on? 

I should also mention, any future submission would explicitly state that this assumption is incorrect, but I would rather address point this directly with the referee, if it can be done so respectfully.
This is a waste of your time, and, more importantly, a waste of the time of people who already selflessly gave away their time working to give you feedback that helps you improve your paper, and to help the peer-review system function.

The best thing you can do is not ask the editor and referees to spend any more time on your paper, unless you are contesting the rejection decision in good faith. And use their feedback to improve your paper so that other referees and readers don’t reach the same erroneous conclusion that this referee did.