Mostly it can be said that fate is responsible for their births into the particular families in wihch they were born into. Fate is responsible for the fact that Romeo was born a Montague, and likewise, that Juliet was born a Capulet. Therefore fate is also responsible for the fact that they are caught in the middle of two feuding families, which ultimately leads to their deaths.
However, choice is responsible for the continuation and bloodiness of the feud for so many generations. We see from the Prologue that the two families have been waring for many generations through the words "ancient grudge" found in the lines,
From ancient grudge break to new mutiny,
Where civil blood makes civil hands unclean. (3-4)
We also see that even the two heads of the households, especially Lord Capulet, acknowledge that they could be making the better decision to maintain peace, as we see in Capulet's lines to Paris,
'Tis not hard, I think,
For men so old as we to keep the peace. (I.ii.2-3)
Fate may also be responsible for Romeo and Juliet falling in love, but choice is responsible for what they did with that love and what ultimately became of them. Juliet is responsible for the choice of marrying Romeo in secret and of faking her death, which ultimately leads to her real death. Romeo is responsible for the choice of killing Tybalt in revenge for Mercuto's death, which ultimately leads to his own death, and Romeo is responsible for his own suicide.