In a meeting yesterday, one coworker was unproductive and, for lack of a better word, sulky. The issue did not get resolved. Today, he sent an email to the team apologizing for his unprofessional behavior and listing some more helpful comments along with some suggestions for moving forward. Do I thank him for owning up to his behavior? Or would that just call more attention to it?
I think that you respond to the apology by focusing on the positive content, which was the useful suggestions for how to move forward. You can choose to do this either in an email to the team, or privately. If you do it privately, you can also add a statement that you appreciated his apology.
Doing this costs you nothing, and it buys you goodwill from him. People generally like having useful suggestions acknowledged. Additionally, acknowledging an apology lets him know that you're willing to excuse an instance of poor behavior when he's identified it and tried to make amends. From all appearances, his email was a highly professional way of handling a situation that he had (likely inadvertently) made awkward and uncomfortable, and responding to his apology recognizes that.
Assuming that "coworker" means that this person doesn't report to you and "team" implies a group of equals, I wouldn't reply to the apology. If you'd like to discuss some of the suggestions for moving forward, by all means reply for that purpose.
The apology is to the team. Having every member of the team potentially coming forward to discuss the situation isn't likely to be helpful-- the person will end up being reminded of the issue a dozen times. It doesn't sound like what the person did is worth that level of discussion and analysis-- he was having a bad day and it showed in a meeting.
If you were this person's supervisor, then it may make sense to have a one-on-one meeting to discuss the issue depending on the person's track record. I'd tend to expect that someone that apologized the next day after merely being sulky and unproductive the day before was just having a bad day and that it's unlikely to be something that really needs a manager's intervention. If it is a pattern or if the behavior was more egregious than what you seem to be describing, then a conversation would be in order.
Talk to him face to face when you are alone.
Just tell him that you have seen his effort, and that it is appreciated, but he has to try to be more focused on what he has to do when he is in a meeting.
Don't insist. Just tell him (if you can be alone together).
(That's my point of view, do what you want. I can give you some advice but you master your life :) )