I think as modern readers we do not see that tragedy is the only outcome, but if you put in the context of when the play was written it is the outcome that the audience would expect.
Marriage, particularly for the nobility, was primarily for alliances or money. Children were seen as possessions, and parents essentially 'owned' them. Any marriage they entered into without their parent's knowledge was not a legal marriage and could be voided by the church. Now add in the views on women, and Juliet's actions are almost inconceivable for that time period. Her complete rebellion and disobedience to both society and her parents can only lead to her own destruction.
To bring it into the 21st century, imagine a young girl who refuses to obey her parents and the law and does whatever she wants - she refuses help and tries to go it alone. Would we expect the outcome to be happy or tragic?
Juliet's refusal to bow to her father only adds "fuel to the fire." Her father is very domineering and doesn't like to be crossed. He tells Juliet she will not be acknowledged as his daughter if she refuses to marry Paris. Her mother is not a very tender,caring or nurturing person. She is only interested in social status and doing what is best for the family image. It is obvious that Juliet's refusal will ostracize her family and lead her to drastic measures so she can be with Romeo.
Well, hmm. Honestly, I don't think it does.
However, if I had to argue that her actions have this effect, I'd say they have this effect for the following reasons:
1) She's a child. Her parents know what's best for her; she's young, inexperienced, etc. This means she's less likely to make good decisions.
2) Her parents have been searching for a good husband for her for some time. Her love for Romeo is instant. This means it is less balanced, and more likely to go astray.
3) If she agreed with them, love would align with love. (She'd marry who they choose, and everyone would eventually be happy.) Instead, love must fight love, and that's always tragic.