Naruto doesn't need closure. He isn't acting like he even needs any, he's beaming. He watched how Sakura went from suffering because she couldn't stop Sasuke from leaving the village, to Sasuke apologizing and taking responsibility for the trouble he caused her. That's Sakura getting closure for the thing that was the hugest failure of her life.
It's pretty apparent by now that Naruto didn't make her a promise with any agenda behind it, like some shippers like to think. Can they not comprehend that sometimes people just mean what they say? I promise I'll bring Sasuke back (so I can give myself permission to confess because this totally seems like an opportunity)? No, it was: I promise I'll bring Sasuke back (because I can see that it's killing you, and that kills me).