Wow, tough call, so I approached my husband for a male perspective. His point of view the best thing and only thing you can really do is try to get both in the same place at the same time. If at that point they still won't talk then each will have to deal with the following consequences. Not very comforting, I know. From what you posted, not a probable situation either.
One thing to keep in mind, and I know this isn't always the easiest thing, as his wife approach him when things are quiet and it is just the two of you. Snuggle up and say quietly, I love you and I want to support you. I know that you are angry with your father. I hope you understand that it is because I love you so much that I have been pushing towards you and your father working out the issue or at the very least setting the issue on the back burning during this medical crisis. Also because I love you, I won't push the issue anymore. I want to know what I can do to support you through this time and I would like for us to make sure that we protect our daughter from all of this internal family angst. I hope that you understand even as I want to support you, because of this medical crisis I do not want to feel like I need to take sides in order to support you. For me, I want and need to be there to support your father and you mother and the rest of your family as your father goes through surgery, recovery and any further medical care. Sometimes, the role of a good wife is to simply step away...and that is super tough!!
One more thing to consider, men communicate completely differently than women. According to my studies in a recent social psychology class, men communicate on a dominate/submissive playing field and there is no in between. When a man apologizes in his own mind, in his own world he becomes the submissive. So if you are talking about a legitimate issue aka one which is pretty darned important to each of them, then by asking them to resolve/resign is the same as asking them to relinquish a bit of their manhood. Of course as a women, I say the situation warrants rising above all of that - but I can begin to fathom admitting being wrong anything other than simply being a human.
Best wishes...and please, please let us know how everything goes!