To what extent is the statement true that Shakespeare's "Romeo and Juliet is a product of the characters' destinies rather than their self-determined actions"?
While fate certainly does have a role in the play, Shakespeare also makes it very evident that the play is more about choices and consequences.
Fate plays a role only insomuch as the characters believe in fate. It is especially Romeo and Juliet who speak of fate. For example, when Romeo's friends try to persuade him to crash the Capulets' ball, he responds by saying that he feels it unwise to go because he had a dream he interpreted as prophesying his ill-fate, his young death, as a consequence of crashing the ball, as we see in the lines:
... for 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, clos'd in my breast. (I.iv.113-17)
Since stars are a reference to astrology and prophecy of destiny, we know that this line is a reference to fate and that Romeo believes his destiny is completely controlled by fate.
However, Shakespeare makes it evident it is not truly fate that is controlling the consequences; rather it is personal choices that are resulting in consequences. Shakespeare makes the importance of choice evident in the very first few lines of the prologue. In the first prologue, not only do we learn that the Capulets and Montagues are battling an "ancient grudge," meaning longstanding feud, we also learn that there had been a period of peace, but that the current generation of Capulets and Montagues have decided to rehash the old argument, as we see in the line, "... From Ancient grudge break to new mutiny" (First Prologue, 3). Hence we see from this one line that it is truly the choices of Lords Capulet and Montague that are responsible for all of the violence in the play, which leads to all of the consequences in the play. Prince Escalus even confirms this viewpoint when he rightly lays the blame for all deaths in the play, especially Romeo's and Juliet's deaths, on Lords Capulet and Montague, as we see in his lines, "See what a scourge is laid upon your hate, / That heaven finds means to kill your joys with love!" (V.iii.303-04). Hence we see that while the characters believe in fate and destiny, Shakespeare shows that personal choices are to blame for the play's tragic end.