When Mercutio and Tybalt duel, Romeo tries to break it up. He steps between the two fighters, and "Tybalt stabs Mercutio" under Romeo's arm (according to stage direction). When Benvolio helps Mercutio to leave the stage, Romeo says of him;
This gentleman, the Prince's near ally,
My very friend, hath got this mortal hurt
In my behalf. My reputation stained
With Tybalt's slander—Tybalt that an hour
Hath been my cousin! O sweet Juliet,
Thy beauty hath made me effeminate
And in my temper softened valor's steel. (3.1.114–120)
Romeo is aware that Tybalt was really coming after him, and Mercutio received his mortal injury while fighting to defend Romeo's honor. Mercutio felt that Tybalt was speaking disrespectfully to and about Romeo, and so he stepped in to fight Tybalt when Romeo continued to refuse. Now, Romeo seems to feel responsible for Mercutio's death, and this sense of guilt and responsibility likely helps to compel him to fight back.
When Romeo sees Tybalt again, he says of him, "Alive in triumph, and Mercutio slain!" (3.1.127). He cannot allow Tybalt to remain living while his friend Mercutio is dead. He says to his enemy:
Is but a little way above our heads,
Staying for thine to keep him company.
Either thou or I, or both, must go with him. (3.1.131–134)
Because Romeo and Tybalt are responsible for the death of Mercutio, Romeo says that one of their souls must accompany his to the next life.