The situation: I had been involved for several years in a group my child participates in and I was put in charge of a subgroup. Realizing I needed more adult assistance, I asked for help from an acquaintance who is the parent of one of my child's school friends who is also involved.
Unfortunately, it turned out that this other adult and I couldn't agree on anything. At every single event the other adult criticized me in front of the youth. He also often usurped authority and took over running the events, even though it became apparent that he didn't know what he was doing. At other times he failed to attend, even though he had promised to be there. Outside of our events he would criticize me for things not going well. He also failed to complete required paperwork, but accused me of nagging him when I reminded him that it needed to be done.
On multiple occasions I tried to discuss things with him outside the events, both face-to-face and by email. He always denied that his behavior was inappropriate and pointed the finger at me for all our interpersonal problems as well as the problems in our subgroup.
Hindsight being 20/20, I should have told this person that his "assistance" was no longer desired at an early stage. However, I continued to let him participate. This led to ongoing anger and bickering between us during events, and eventually a "blow up" in the middle of an event, when I finally told him that I didn't want his help any more. Fortunately, the youth participants were otherwise occupied and were unaware of what happened.
I realize my reactions to his behavior were poor. Generally, I think it is best to apologize when that's the case. However, in this circumstance I find it impossible to formulate an apology that would be sincere without saying something like "I am sorry for my reactions to your poor behavior." - which would probably not be well received.
So, how should one apologize in a situation like this?
Edit: The events described above actually happened a few months ago. Initially I made a few gestures to improve the situation without apologizing (e.g. sending memorabilia from our events to him, trying to say "Hi" when we crossed paths). At this point, the other person won't even acknowledge me, even in situations where that makes things extremely awkward. I have no interest in a friendship with this person and there is no chance that we will be volunteering together in this group in the future. While there is a bit of a "I should apologize" thought in my head, there is also the expectation that he would try to use an apology against me (he filed a complaint against me with the group after the "blow up" - the finding of the investigation was - as I initially stated - we were both in the wrong). The main practical reason for an apology would be to avoid this awkwardness, since we are sure to continue crossing paths for at least the next couple years (while our children are at the same school).