My boyfriend tells an ex-friend of mine my personal business and I'm going to ask him not to.
What has led you to think you should tell him that?
I don't like that he tells that ex-friend very sensitive information about me
That definitely seems like he's oversharing!  I think it's fine for you to expect that he doesn't tell all these personal details to people you don't want knowing them.  I hope he listens to you!
Unfortunately, I told him and we got into an argument and now he's ignoring you.
It's unfortunate that your boyfriend reacted negatively because he's wrong for sharing your private information with your ex-friend against your wishes. I would let him calm down and address the issue again and let him know you are serious about your request to respect your privacy.