It can be said that Romeo and Juliet are only partially responsible for their own deaths. While it is true that their own youthful choices precipitated their deaths, their families' feud was the ultimate cause.
In the final scene, Prince Escalus rightly places the blame for Romeo's and Juliet's deaths on the longstanding hatred the two families have felt for each other, as we see in his lines,
Where be these enemies? Capulet, Montage[Montague],
See what a scourge is laid upon your hate,
That heaven finds means to kill your joys with love! (V.iii.302-204)
In pointing out that God has chosen to punish their perpetual hatred with the deaths of their children, Prince Escalus is also laying the blame of their deaths on Lords Capulet and Montague. Had the two families not felt animosity towards each other, the couple would have been free to marry in peace and Tybalt would not have challenged Romeo, which leads to his own death, Mercutio's, Romeo's exile, and indirectly, Romeo's own death, as well as Juliet's.
However, even though perpetual hatred is the ultimate cause of both Romeo's and Juliet's deaths, we must remember that both characters are very, very young. As a result of youth, they both make some very unwise, emotionally driven, and rash decisions. The couple could have made the rational decision to hold off on marrying until they had let their love for each other be made known to their families. It is not a stretch of the imagination to believe that Romeo might have been able to earn Lord Capulet's respect. We learn at the ball in the first act that Capulet, along with all of Verona, acknowledges Romeo to be a "virtuous and well-governed youth" (I.v.71). Since we know that Romeo is generally respected and thought well of, we may be able to assume that he can earn Capulet's respect, who might then consent to the marriage. Juliet also makes a rash decision in agreeing to fake her own death, rather than confess to her father about her earlier marriage, through the friar's help. Finally, Romeo acts rashly when, after seeing Juliet in the tomb, he still believes her to be dead, especially after noticing that she still has color in her face. Only someone who is very young and inexperienced would assume that she is truly dead and that she still has color in her face simply because she is beautiful. We see him observe the color that has returned to her face in his lines,
Death, that suck'd the honey of thy breath,
Hath had no power yet upon thy beauty.
Thou art not conquer'd. Beauty's ensign yet
Is crimson in thy lips and in thy cheeks,
And death's pale flag is not advanced there. (V.iii.92-96)
Because her lips and cheeks are still red, we can see how young and naive Romeo truly is to continue to believe that she is dead. Had he realized she was still alive, he would have spared both his own life and Juliet's.
Hence, while hatred is the primary cause of their deaths, youth and naivety also precipitated their deaths.