As an individual who suffers from moderate/severe anxiety, along with a couple other disorders. I for one can say that I am always worried that my illnesses will cause even the closest people around me to finally say "ENOUGH!".
What sets me apart from everyone else though, is that i've been in both of you and your friend's shoes. No matter how many time your friend apologizes and no matter how many time you say "It's ok", "Don't Worry, or "I get it". Your friend will never stop apologizing, not because he didn't believe you the first 20,000 times but for the reason that it's his coping mechanism for dealing with the anxiety. While also being a part of the anxiety itself.
He needs to be reassured that you "Forgive Him" to help him feel a sense of calmness, even though he knows you couldn't care less about judging him.
The best way that I've found to deal with type of anxiety is to slowly turn these types of 'serious' moments into 'jokingly laughable' moments. Start by kidding around when you answer him. Throw a "I can't forgive you", "No Way, its not alright", "You better be sorry!" once in a while. Start slow and continue to slowly increase the rate at which you foolishly answer. ALWAYS make sure that your friend knows that you are kidding, by means of laughter to even saying "I'm only kidding". The most important part is that they know you are only joking.
Soon enough both you and your friend will become way more comfortable and hopefully closer with one another. Although, i wouldn't count on any "reassuring habit" to disappear completely.
Good luck to both of you, and your friend is very lucky to have somebody like you that they are able to call a friend.