While some might argue that the main antagonist is either fate or the Capulet and Montague families, I actually argue that the main antagonists are hatred and Tybalt.
The reason why hatred serves as the antagonist instead of the two opposing families is that hatred between the two families is the cause of the protagonists' deaths rather than any direct actions of the families, especially the heads of the families. When we see Lord Capulet stop Tybalt from attacking Romeo at his feast, we learn that Lord Capulet would not actually ever want to directly harm Romeo, as all of Verona thinks he is a "virtuous and well-govern'd youth" (I.v.71). Because of this we know that the hatred the two families feel for each other does not have any concrete foundation but is rather hatred for hatred's sake. Even the Prince blames their actions on their hatred, referring to the families as
You men, you beasts,
That quench the fire of your pernicious rage
With purple fountains issuing from your veins! (79-81)
The phrase "pernicious rage" refers to wicked, violent fury (Random House Dictionary) while the reference to "purple fountains" refers to the families' bloodletting. Hence, we see through this line that the Prince blames their actions on their violent hatred for each other. We see it even further in the final seen when he declares, "Capulet, Montage[Montague], / See what a scourge is laid upon your hate" (V.iii.302-303).
Tybalt can also be seen as the primary antagonist because it is his own hotheaded anger that indirectly causes both Romeo's and Juliet's deaths, as well as many of the fights that take place. While the Capulet and Montague servants began the fight in the opening scene, Benvolio tries to put a stop to it while Tybalt makes it worse. Tybalt begins fighting Benvolio, which leads to Lords Capulet and Montague joining in, which further leads to the full city riot. Likewise, Tybalt's anger leads him to revenge himself on Romeo for crashing Lord Capulet's feast, which directly leads to both Tybalt's own death as well as Mercutio's and eventually leads indirectly to both Romeo's and Juliet's deaths, due to Romeo's banishment. Hence we see that because Tybalt is primarily responsible for the play's grief, Tybalt can also be considered the primary antagonist.