Well, Naruto knew for a fact that Obito was genuinely sorry and felt miserable for his crimes. And Obito wanted to die. He didn't ask for forgiveness. Coupled with his final actions, that is why Naruto forgave him. It won't be as simple with other people, as Obito and Naruto were pretty much the same person.
Also, he's battling the Curse of Hatred. Can't really fight it well if you can't forgive your greatest enemy when he's truly sorry.
On top of that, Tsunade sorta did the same thing. She left Sasuke alone even though he was working for Orochimaru and later Akatsuki. I know it was because of Naruto, but still.
And finally, he'll have Shikamaru as an advisor, so things can't be too bad with someone so intelligent as second in command.