This is a really interesting question raised in light of the drama. If we asked the protagonists, they would probably say that they had no choice but to die for their love. I think that this probably is a bit on the exaggerated side. There were plenty of opportunities present where choices were made. Both Romeo and Juliet made distinct choices in regards to their own senses of self and of one another that had results. The idea of destiny is a Romantic one in the play, and helps to create a great sense of drama. Yet, I do believe that they did possess a sense of autonomy and choice and deliberately took paths that resulted in specific outcomes. On a larger level, I would say that individuals do possess freedom and choice. There might be a larger design that lies outside the scope of human comprehension, yet this does not entirely absolve individuals of actions and freedom in this scope. Even the mere notion in believing in destiny is one of choice, as one chooses to believe in such an element and proving that freedom is something that individuals possess.
I do not in any way believe that people have destinies that cannot be avoided. Of course, there is no way to prove this one way or the other.
Now could Romeo and Juliet have chosen differently? Of course they could have (except for the fact that Shakespeare was controlling them and they had to do what he wrote.). I mean, nothing said that they had to keep trying to see each other after they had found out they were from enemy families. Nothing said that they had to sneak off and get married. They may talk about fate, but I think they are just using that as an excuse to do what they want.
Of course, the story would be way different if they hadn't and it would not be famous. But it wasn't destiny, in my opinion, it was their own impulsiveness.