In my opinion, this duel is the fight of all fights to put an end to who really gets Juliet. Obviously, previous to this moment, Romeo had her heart, and Paris had the right to have her according to the culture of the era. The winner of the fight figuratively has Juliet for all eternity since they both believe she is already dead.
However, this fight performs the literal function in the story of letting the two guys working for a woman's heart to meet. It also demonstrates the impulsive desperation that Romeo is willing to go to. Finally, this fight serves the purpose of drawing toward a final climax in which we will discover that Juliet is not really dead. The fact that neither of them know Juliet is alive also performs the purpose of dramatic irony. This occurs when we the audience know something that characters do not know.
I think it begins to prove that Romeo is not in a correct state of mind and is just acting on his impulses.