Scotticher, I also believe that the word “but” has no place in an apology. The apologizer doesn’t have to lie and say he agrees with you, but there’s no reason for him to use the apology to reiterate his opinion. In my case, the guy should have just said “I’m sorry” and left it at that, without getting in another jab.
Someone told me, “Anything preceding the word ‘but’ in a sentence is a lie”. Now it drives me batty.
That drives me nuts as well. My husband had a terrible tendency to do that. Or, I should say, he did that on the rare occasions when the moon was in the seventh house, Jupiter aligned with Mars, and something that tries to resemble an apology can be magickally summoned forth from his lips.
Drove me so crazy that once, not too long ago, I just interrupted him as soon as I heard that “b” sound after the “I’m sorry.” “But nothing,” I said. “Either accept the fact that you are actually in the wrong on this issue, and a real apology is owed, or don’t bother.”
To my utter amazement, that seemed to work. Apologies are still really, really rare, but he doesn’t follow them up with a “but” anymore. 
Once again, I don’t agree. Seems like an awful lot of weight to place on a tiny little word. How you say something may or may NOT mean something.
Now in Persephone’s case, with a long relationship behind it, I would be very inclined to accept that she knows whereof she speaks concerning what her husband means.
But to say that EVERY time someone uses “BUT” in an apology negates it? I don’t think so.
How about "I still don’t agree with you, BUT I am sorry for the way I dealt with this situation. "
How about “I don’t think you are dealing with this the way I personally think you SHOULD be dealing with this, BUT I love you and support you.”
How about giving someone the benefit of the doubt?
Seems quite straightforward to me, with no “but” expressed or implied.
JMHO, BUT…I think I am right. 
Scotticher, I still think there’s no reason to reiterate your position. It doesn’t belong in an apology. Just say you’re sorry and stop trying to continue the argument. This guy told me I was “silly,” whereas before he had called me “insane.” I don’t see a huge improvement. Come to think of it, a friend can call you silly, but if a stranger does it’s pretty insulting.
Well, you actually read the “apology” and I didn’t, so I’m sure you have a better understanding as to how sincere it was. I still think it is in your best interest to accept it, but I understand if you don’t feel like doing so.
Oh, and I apologize to Pers…I didn’t word THAT very well. What I meant to say is that in a case LIKE hers, I would be inclined to accept…and so forth. In HER actual case, I have NO doubt that she knows exactly what her husband means by his “apologies.”
An apology is ‘An admission of error or discourtesy accompanied by an expression of regret’.
Getting in a final jab with the apology is a bit tacky, and devalues it a little, but in some circumstances it’s needed to clarify that although you’re sorry, the situation has not changed, which can be unclear if you just say “I’m sorry”. However it really should be done tactfully. An apology does not need to indicate a change of opinion, it just needs to let you know that they realise that whatever they did could have been done better, or didn’t need to be done at all.
If this guy has said, “I’m sorry for what I did, my behaviour was inappropriate, but I think I’m right”
I would then respond with “Your apology for your conduct has been accepted.”
He’s apologised for what he did, not what he thinks, they’re seperate things as are the apology and the continued opinion.
Likewise this allows you to accept his apology for what he did, but not forgive the reasons why he did it.
It sure doesn’t sound like an apology to me, especially if he called you ‘silly’ right in the middle of it. One can say that they still think you’re wrong, and that their actions were inappropriate, but you have to stop with the insults.
Scotticher, I think the problem is that the “but” seems to continue the argument while claiming to be an apology. If you said something and hurt someone else in the process, you don’t have to lie and disclaim it. You can simply say, “I obviously hurt you by how I phrased that/my vehement defense of my opinion/etc., and I’m very sorry, I didn’t mean to do that.” This does not back down from your position, but expresses regret.
I once read that there are three components necessary for an apology, and that the apology shouldn’t include too much more or it usually ceases to be one (based on what else you add, like ways to continue the dispute):
- Say you’re sorry
- Say you didn’t intend to cause pain
- Say it won’t happen again
In chula’s case, the apologizer could have left it at something like “I’m sorry I insulted you during our debate, that was uncalled for”, rather than absolutely having to get in a “but I’m still right” comment as well. In no way does an apology for an insult imply that their original position was wrong.