Mercutio is Romeo's fun-loving friend who loves to play with words and delivers some raunchy humor early in Romeo and Juliet. However, when Tybalt, Juliet's cousin, challenges Romeo to a fight and Romeo declines (How can he fight his new secret bride's cousin, after all? Not the best way to begin a marriage), it is his buddy Mercutio who steps up to defend Romeo's honor.
Mercutio is killed and in dying, he curses both the Montagues and the Capulets for their ongoing animosity that has resulted in his death. So you could argue that in that way, he's a bit responsible for the fallout of Romeo and Juliet's deaths. (This would assume, of course, that the curse holds some actual power that plays out over the rest of the play.)
In response, Romeo now feels that he must fight Tybalt after all. He notes,
This day’s black fate on more days doth depend.
This but begins the woe others must end (III.i.81-82).
Romeo understands that because of Mercutio's death, there will be more death. He then kills Tybalt, and that gets him banished. And his banishment sets up the circumstances necessary for both his and Juliet's deaths.
For the most part, Mercutio doesn't directly cause the deaths of Romeo and Juliet, but his choice to fight Tybalt certainly changes the course of their lives.