You seem to want to argue with your H about how much you do care. Really you can't. Perception is 99% accurate. If he says that's how he sees it, then that's how it went down for him. The best you can do is say, I disagree, but I think I can understand how you saw it that way. To avoid misinterpretations in the future, let's be clear from here on out, I do care about your health.
And drop it completely.
Ask yourself before everything you say from now on: Do I want to be right, or do I want to be happy? Is this the hill I really want to die on?