The best fighter certainly cannot be based on if he was defeated or not. I mean, look at
Ken Jennings from Jeopardy. So just because some are undestroyable such as the King of the Dead, doesn't make them the best fighters. After all, he who is highest falls hardest. So what does that mean? People with more power, such as Gandalf, Sauron and the Witch King, have much greater weaknesses.
But if it were to come down to hand-to-hand combat, battle to the death in a sumo ring, then there are a few things to consider.
Stamina: Obviously Elves and Dwarves would last longer in a physical fight then men and wizards would. They're built for it.
Strength: What race is strongest, again, toss up between Elves and Dwarves.
Agility: Jeez, this isn't really leaving a lot of room for men is it?
So I've come to the conclusion that if they were to have an All-Middle-Earth wrestling tournament, Elves and Dwarves would probably kick bum because of how they're built and because they're so much older then men. I think that Gandalf wouldn't fare well in such a showdown if he couldn't use supernatural powers. Yes, he's a good fighter, he's lived a long time but he has the misfortune of being trapped in the body of a decaying man. Not good for a lasting opponent.
So who's the best fighter? In terms of Third Age type warriors, I'm tempted to say Aragorn only because I can see his spirit and skill beating most of the other candidates. He'd find a way...and that's very important. But then I wonder if her could take on Legolas? Or Gimli, or Elrond for that matter!