Julie, I apologized to my H. I did it for me, because I felt so awful. Like you, I took a good hard look at my own behavior and was really upset with myself for things I did that led us to this point. He bears some responsibility, sure; but, my part in our friendship dying was not insignificant. I was critical, cranky, irritated with him because I thought he should be able to read my mind. ML became a rarity, and it is something he craves to feel good about himself.
I just told him I'd been thinking about things. I clearly saw my part in our M falling apart. I never meant to hurt him or disappoint him. If I had it to do all over again, I would make different choices.
He didn't fall over in gratitude, and it didn't change our situation. My personal belief is that if I've harmed someone in any way, I need to accept responsibility, apologize, and make amends if possible. H won't let me make amends right now, but who knows what the future will bring? My soul settled down dramatically after I apologized.
If the only thing you do is offer your H a sincere apology, I don't believe that's considered pursuit.