Michael needed to hear it himself. If you remember in the novel, in that very scene, it was implied that Michael could never be the man his father was, because Vito wouldn't have needed to hear an admission. What he knew in his heart would have been enough.
It would have been easier for Michael to avenge Santino because Michael was away in Sicily all these years, and never really sat down to dinner with Carlo. It was always Vito, Mama, Santino, Tom, Connie, Carlo, and the kids at breakfast, lunch, and dinner. A real affection was there, you don't break bread with someone you have meals with every day for years and years and then cold-bloodedly kill them. That was Vito's problem.
I also noticed that Mama Corleone and Carlo had a very deep relationship. During the wedding sequence, when Johnny Fontaine starts to sing, you can see Carlo and Mama holding hands. Almost like they were the newlyweds. (It was real, and doubt it was intentional for the film - I think it was Morgana King and Giavani Russo holding hands, not Carlo Rizzi and Mama Corleone, but it was shot in the film regardless.)
Michael was always detatched from family functions, and then he was exiled to Sicily for many years, so killing Carlo probably wasn't that hard for him to do.
