Nope. An order does not have to be in writing. It would certainly be nice, because you might want to document the fact that you were given a questionable order, but nobody is required to actually give one.
As for the subject of throwing rocks… The law requires the soldier to use force to counter a perceived threat, and a rock can certainly inflict a great deal of injury. It very much depends on the situation. If you see a bunch of Iraqi kids throwing pebbles, nobody is going to take that very seriously. And IIRC, there was a very notable incident in Kosovo in which protestors hit a soldier in the face with a brick, but the soldiers exercised restraint and did not open fire.
That said, go watch some Youtube videos of Iraqis throwing RKG’s at American vehicles. How easy do you think it is for someone to tell the difference between a rock and a hand grenade? If I were a bad guy, I would have kids throw rocks and shoes all day long in order to desensitize the soldiers, and then once they let their guard down I’d toss a hand grenade into the mix. If I was a bad guy.
I was in Iraq one time, where the kids in a local village routinely threw rocks at us. We were certain that the adults in the village were inciting them to do it, on the understanding that Americans would never shoot innocent children. Our Platoon Leader eventually took aside the village elder and told him, “We understand they’re just kids and we’re trying to be patient, but grenade attacks are on the rise and I don’t want there to be any misunderstandings.” They took the hint.