Spartan vs. Ninja
Their decision: Spartan
My decision (1v1): it depends. Spartan (absolutely) in a straight fight, Ninja (absolutely) in a real fight.
My decision (war): Spartan
This is an odd matchup bordering on stupidity. Ninja were assassins, not straight up warriors. This means that they didn't kill when the opponent was right in front of them. They attacked when the enemy didn't know they were there. If put in a straight fight, they can't win. Their weapons are built for either escape, injury, or murder. These aren't useful at all in a 1v1 duel. The spartan, on the other hand, was the perfect dueler. His weapons don't mess around. His shield is massive, tough, and packed a heavy punch. In a straight up fight, it's no contest. The spartan would annihilate the ninja.
However, if we take the ninja's weapons and tactics seriously, he would probably win in most situations of real life. That blowgun coupled with stealth is enough. If the spartan has no idea that the ninja is there, he'd be dead in one shot. Ninjas were murderers. They were the best at what they do. One could easily hide on top of a tree or something and aim for any exposed part of the spartan's armor, or even wait until he goes to sleep, and that would be the end of it, even before the fight begins.
In a war, however, the spartan phalanx wouldn't fall for such a trick. Interlocking shields would be able to protect the majority and when the fight gets close, it will be simple destruction. Ninjas have no chance at all.