Suppose Naruto were a few seconds too late and Sakura had died in his arms. How would Naruto have reacted to Sasuke, per the way he had been written around this time and the remainder of the manga? My honest sincere belief: No jokes and not even kidding . . . Naruto is upset about it . . . for a total of two minutes . . . maybe three . . . no more than four . . . before pinning the blame on the matters allegedly beyond Sasuke's control (i.e. cycle of hatred). The remainder of the story then proceeds as normal without incident.
There are many different levels to this, so I will separate my response in them.
Theme: I doubt forgiveness/redemption is the primary theme or, at least, in the form we saw it. Most of the audience would never accept it and the hope is that someone writing it would see that. That said, its these kinds of tests that I wish the manga did. One of the reasons I hate the end is that Naruto's answer and belief system are never really tested, so its hard for me to get behind it. The closest is Pain, but Naruto is bailed out from the consequences by Gedo Mazo. The Third's hesitation on Orochimaru had tragic consequences. Naruto redeems/forgives some terrible people, but it never nukes in his face. His plot armor always protects him. Anyways, I think the theme is totally different or we get a much grayer if not totally dark view of these themes, which almost everything that follows.
Would Naruto forgive Sasuke? It would probably look something like with Pain because of the how much value Naruto places on his ideals, but Sasuke either is a permanent exile or a dead body at the time it happens. I mean, Naruto allowed Orochimaru and Kabuto to remain free despite their crimes, the legendary Obito line, going to bat for Sasuke and so on.... Obito killed Neji to prove a point and still Naruto stopped Kakashi from killing him. So if he doesn't kill him in a fit a rage (certaintly possible that Naruto does do this), then yes, Naruto probably eventually does, but Sasuke never returns or Sasuke is dead when he is redeemed.
Collateral Damage: Would he go 6 tails like with Pain? I think he probably would because Sakura is one of the closest people too him and the cause of her death would be right there. Might he be able to hold back some of his anger because its Sasuke? I don't know. That sense of betrayal might make things worse. If he does, Sasuke, Kakashi, and Karin are probably going to die. If they don't, its because Obito intervenes and gets his 9 tails chakra much sooner and probably puts Naruto out of commission (I admit I cannot remember what Obito was doing at that monent). Moon's eye is probably initiated much sooner and probable done without a fourth ninja war, If nothing else the next arc is probably a rescue Naruto arc. Either way, things would probably head south and do so in a hurry for one reason or another.
The fandom: If you thought the fandom wars were ugly before.... Don't forget, Sakura attempted to assassinate him from behind and stopped because of a lack of nerve. There would be an inevitable split between those who though him justified or those that didn't.
In the Boruto manga, Sakura didn't seem to be bothered by the fact that Naruto was in danger. It's probably a spoiler, but there was a sequence in which Hinata put herself in danger to help him (but failed) and there was not a moment in which Sakura was in worry. Probably because she is a professional, probably because she knew he wouldn't fail, but it made the impression as if she didn't care? 
