Since Michael chose to be Don, he had to kill Carlo. Even if he forgave Carlo's treachery personally, it would have shown him to have a "weak spot" that his foes would have exploited. And, as Tom explained to Kay (in the novel), "...treachery can never be forgiven...they [Tessio and Carlo] would have been a danger to all of us, all of our lives."

As Roth said, "This is the business we've chosen." Murder comes with the territory.