You have been civil and decent. If you feel you did anything to apologize for, then apologize. If not, let it go. Whether your brother WANTS an apology is pretty much irrelevant.
I wonder, though, if it isn't your dad that simply assumes that your brother is waiting for an apology.
It really might take some of the pressure off future stresses if your dad were to appoint someone other than either of you as his executor. Unless his estate is complicated, it probably wouldn't take that big a chunk to have an attorney do it.
My ex-husband has made me the executor of HIS estate (with his wife's knowledge and consent) because he trusts me and because, I suspect, he thinks I would be less emotional and better able to deal with the logistics. So even a third party, such as a trusted friend who will be neutral, would be an option.