The inevitability of Romeo's and Juliet's deaths stem from two factors: 1) the family feud that divides them from being together; and 2) the fact that they are both young and make very stubborn and juvenile decisions. Had the characters been older, even with the feud raging between them, they may have still survived.
The fact that the family feud divided and eventually killed the lovers can be traced all throughout the play. It can especially be seen in Tybalt's violent reaction towards seeing Romeo at the Capulet feast. Tybalt's anger, which was fed by the feud, leads to the deaths of Mercutio and himself and Romeo's banishment. Accepting the fact that Romeo was banished as a result of the feud is especially key to understanding that the feud led the deaths of both Romeo and Juliet. Had the feud not existed, had the two families been allowed to intermingle, Tybalt's own fiery, violent temper would not have been fanned by the feud. Hence, Tybalt would not have challenged Romeo, which led to Tybalt's own death, Mercutio's death, and eventually, indirectly, Romeo's death. Shakespeare's decision to rest moral blame on Romeo's and Juliet's deaths can best be seen in one of Prince Escalus's closing speeches:
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.302-304)
In this passage, by referring to "scourge" he is arguing that heaven has decided to punish Capulet and Montague for their continuing feud and hatred of each other by killing that which they both loved.
The fact that both Romeo's and Juliet's youthfulness and decisions also led to their own deaths can also be seen all throughout the play. Their worst decision was to marry so hastily and without parental awareness. Had Lord Capulet been aware that his daughter wanted to marry Romeo, he may not have tried to force her to marry Paris. As Capulet tells Paris, he considers his own consent to the marriage as only a portion necessary for the bargain. He cares a lot about his daughter's own interests and feelings. This is evident in his response to Paris:
But woo her, gentle Paris, get her heart;
My will to her consent is but a part.
An she agree, within her scope of choice
Lies my consent and fair according voice. (I.ii.16-19)
Since we see that Capulet cares about his daughter's feelings and later we learn that Capulet even believes Romeo "to be a virtuous and well-govern'd youth" (I.v.71), it is safe to assume that had Romeo and Juliet been less hasty in their decision to marry without at least parental acknowledgment, their unity would have fared better and their lives may have actually even been saved. We see that their youthful, hasty decision led to their demise expressed in Friar Laurence's statements. One statement he makes to indicate his opinion that the marriage is hasty and foolish is given just before he marries them: "These violent delights have violent ends" (II.vi.9).
Hence we see that Romeo's and Juliet's deaths were inevitable and were caused by both their family feud and their youthful, naive, and hasty decisions.