Revealing mistake: When Rambo throws a rock at the police helicopter and knocks Galt out, you can clearly tell it's a stuntman. Galt is in his 40s and overweight. The guy that was falling looked like he was 25 and was noticeably thinner.
Revealing mistake: In the scene where Rambo is driving the truck and pushes the cop car into path of the broken down car, the explosion starts the second the cars make contact. In the next shot during the fireball, the two cables that held the static vehicle in place are visible.
Revealing mistake: Towards the end of the film where Rambo shoots up the computers in a building and the spent rounds are falling to the ground. It is clear to see that these are blank rounds. Where the bullet would sit in the case normally you can see the case is extended, This is where on a blank it is just crimped closed due to the lack of a bullet.
Chosen answer: No he doesn't kill anyone. He didn't know if he was surrounded or not and if he did encounter one from the side they might have shot him.
Grumpy Scot
Rambo killed Galt, albeit indirectly and unintentionally. In the US, "manslaughter involves causing the death of another person in a manner less culpable than murder." Rambo was responsible for throwing a projectile that struck the helicopter, causing the pilot to lose control resulting in Galt falling to his death.
But if Galt had been buckled in, he'd have lived. He was behaving recklessly. Yes Rambo's rock was involved, but that's two steps removed from Galt's death. Rock thrown -> pilot overreacts -> Galt falls because he wasn't strapped in properly. Galt's own behaviour and the pilot's reaction are more at fault than Rambo. This isn't a court, by most reasonable standards Rambo didn't kill Galt.