How much does fate contribute to Romeo and Juliet meeting in William Shakespeare's play?
Much depends upon the instructor's particular slant when teaching the play, normally to 9th graders. The question of fate comes up in several Shakespearean works, although in Romeo & Juliet we get a definite sense of fate over free will. That is, until we get to the end of the play.
They meet by sheer chance, or fate. Two lovers, seemingly perfect for one another, yet from warring families, can be coincidence. After the initial realization that they are not compatible with one another's families, the Montagues and Capulets, much of the rest of the play relies heavily on Romeo and Juliet's rash decision making.
The balcony and subsequent love scene wherein the two lovers conspire to marry secretly, involving the monk, brings about the issue of free will, and why these young lovers choose the path they do. They freely plot to meet in secret, and the monk freely decides to act in their favor.
Unfortunately, everything that can go awry does, as Juliet is thought to be dead from drinking the draught of sleeping potion from the apothecary. Hence, Romeo, showing his rashness again, leaps to conclusion, thereby inflicting his own death. Fate does not drive Romeo to kill himself, nor does fate compel Juliet to plunge the knife within her chest. Their immature decision making skills lead to no good and ultimately their own demise. Fate does not dictate here.
Romeo and Juliet are so blinded by young love that they make hasty choices that could have been avoided. To say that both characters are controlled by fate is to admit that we, as human beings, take no responsibility for our actions, hasty or otherwise.
So, in conclusion, fate plays his ugly hand in the beginning of the play, but the concept of the lovers' free will or choice is more prevalent in the second half.
The degree to which you believe that fate was involved in the meeting of the two lovers in Romeo and Juliet by William Shakespeare depends, to a large degree, on how much you believe in the notion of fate.
The phrase "a pair of star-cross'd lovers" does suggest a sense of fate, in that the phrase is derived from astrology and indicates that the unfortunate fate of the lovers has been predetermined. This theme of astrological determinism recurs at the end of the play, where just before he commits suicide, Romeo states that he:
Will ... shake the yoke of inauspicious stars
From this world-wearied flesh" (5.3.111-112).
Within Shakespeare's religious world, the heavens or stars are closer to God than the earth, and the stars mirror and indicate the will of God. Although God has divine foreknowledge, and thus the stars can make accurate predictions if read correctly, humans still have free will and can make moral choices. Thus from God's point of view (to which humans have no access), everything is ordained by fate, but from a human point of view, Romeo chose to attend the ball, a choice resulting in his meeting Juliet.