I don't think it's purely damage, I'm pretty sure it's the number of "points" you score towards the objectives, similar to how the war zones work. Since most of the fleet actions are "kill X ships of this type" it ends up being mainly a damage thing, but only indirectly. That said, I've never actually come in 1st, but I don't play tac officers, so that could have something to do with it.
On a related note: are the places by team, or individual? I've noticed in my 2nd and 3rd place finishes that the entire team gets the same rank, is it the same for 1st?