What is the role of destiny in Shakespeare's Romeo and Juliet?
As early as the Prologue, the audience learns that the lovers have sprung from the "fatal loins of [...] two foes"; fatal, here, means fateful: thus, they are all involved in a destiny from which they cannot deviate. Further, Romeo and Juliet are described as "star-crossed lovers," meaning that they are destined to be terribly unlucky in love (6).
Romeo is the first of the pair to have some inkling of this. On the night of the Capulets' party, as he has just become convinced to accompany his friends, he says,
my mind misgives
Some consequence yet hanging in the stars
Shall bitterly begin his fearful date
With this night's revels, and expire the term
Of a despised life closed in my breast
By some vile forfeit of untimely death. (1.4.113-118)
He seems to intuitively understand that, by going to the party, he will set into motion some kind of fated plan; again the reference to the stars indicates destiny: something predetermined. This party initiates his fate, and he prophesies that it will end with his too-soon death. Such ability to prophesy is most often connected with the idea that we each have a destiny which we cannot help but fulfill.
After slaying Tybalt in rage over Tybalt's slaying of his friend, Mercutio, Romeo cries, "O, I am Fortune's fool!" (3.1.142). By this, he means that he feels like the plaything, or something meant to entertain, the goddess Fortuna who controls the fate or destiny of humans by spinning her wheel.
Later, when Romeo leaves Juliet's bedroom, she says, "O God, I have an ill-divining soul! / Methinks I see thee, now thou art so low, / As one dead in the bottom of a tomb" (3.5.54-56). In this moment, Juliet believes her soul has the ability to prophesy evil and that she is glimpsing something of their future. Indeed, she is. She will, later, wake up in her tomb to find Romeo dead beside her. Such ability implies that destiny is in charge, despite all the lovers' attempts to thwart it.
Therefore, one could make the argument that destiny (or fate) plays a significant role in this play. Certainly, if Romeo and Juliet were able to make plans or decisions not frustrated by events that were totally out of their control, it would be more difficult to make such an argument, but they are, clearly, met, time and time again, with awful, terrible luck. The fact that they can only really be together in death seems to signify that "the stars" would simply never let them be together in life.