Fate plays a major role in the outcome of Romeo and Juliet through timing--or, more specifically, through the play's characters being at the right (or wrong) place at the right (or wrong) time.
We first see this in the streets of Verona when a random messenger stops Romeo and his friends in order to seek help with reading a document he is supposed to deliver. Through this chance meeting--arguably an act of fate--Romeo learns of the Capulet's ball and decides to attend it in hopes of seeing Rosaline. This, of course, leads to him meeting his lover, Juliet, instead. Whether or not this love is a matter of choice or fate remains to be seen, but the meeting itself was certainly all due to the universe's timing.
Fate strikes again in the form of a messenger after Romeo is banned from Verona for killing Tybalt; this time, however, fate is not so kind. After Friar Laurence's messenger fails to deliver the news of Juliet's plan to fake her death, Romeo believes that his love has actually perished and commits suicide in the Capulet crypt. Juliet wakes to discover her dead husband and consequently kills herself as well. This is an example of being at the wrong place at the wrong time.
Clearly the lovers were destined to meet, as well as destined to part. Their ultimate demise harkens to the opening of the play, in which the prologue declares that they are "[a] pair of star-cross'd lovers." Ultimately, they were only racing against time and their own destinies--a fruitless race and one that resulted in their deaths.