Tybalt, in his bloodthirsty hatred of the Montagues, played a critical role in instigating the sequence of events that led to the deaths of Romeo and Juliet. The critical scene is act 3, scene 1, when Tybalt, after challenging Romeo to a duel, kills Romeo's friend, Mercutio, instead. This act drives Romeo into a rage. The two fight, with Romeo killing Tybalt.

Already at this point in the play, Romeo and Juliet have been secretly married by Friar Laurence, but now Romeo is a criminal, facing banishment for the killing of Tybalt. Meanwhile, Juliet, in her grief, finds herself being pressured by her father to marry Paris (even threatened with disownment should she continue to refuse). Her desperation is so great at this point that she threatens to commit suicide should she be forced to marry Paris. It is in this context that Friar Laurence devises the plan to fake her death (a plan that later goes tragically awry).

In this context, Tybalt certainly holds a lot of responsibility for the deaths of Romeo and Juliet, but I think it would be a simplification to give him sole responsibility. For one thing, you should recognize the role that other characters played in the sequence of events that led to Romeo and Juliet's deaths: Friar Laurence, for example, who married the two in secret and devised the plot for their reunion, not to mention Romeo and Juliet themselves. Furthermore, you should take into account the larger context of vendetta between the two families. While Tybalt, more than anyone else in the play, embodies this history of vendetta, you should keep in mind that the feud is actually far bigger than Tybalt himself. Thus, even as we should assign blame to Tybalt for driving the sequence of events that led to their deaths, we should also blame the larger vendetta and all the Capulets and Montagues collectively who played a part in continuing it.

Tybalt has a clear and potent hatred of everything that the Montagues stand for. His loathing is most obviously displayed during the Capulets' ball. When he notices Romeo, who has gatecrashed the occasion, he immediately wants to confront and remove him. When he informs his uncle, Lord Capulet, of his plan, he is harshly admonished and feels both insulted and humiliated, first by Romeo daring to enter their private celebration and, second, because his uncle scolded him for wanting to do what he believes is right: honoring and defending the Capulets against a despised enemy.

Tybalt is so angered by Romeo's audacious act that he later sends a written challenge to the Montague household in which he wants the family to answer for what he believes was an insult to the Capulet name. Mercutio tells Benvolio that Romeo will answer the challenge but doubts whether he is man enough to stand up to Tybalt.

In Act lll, scene l, after a verbal altercation between Tybalt and Mercutio, Romeo arrives on the scene and is immediately challenged by the former. Romeo refuses to engage him in a duel and says:

I do protest, I never injured thee,
But love thee better than thou canst devise,
Till thou shalt know the reason of my love:
And so, good Capulet--which name I tender
As dearly as my own--be satisfied.  

His response is undoubtedly informed by his affection for Juliet, Tybalt's cousin. The hotheaded Mercutio is upset by what he sees as Romeo's weakness and takes up Tybalt's challenge. Romeo tries to intervene when the two start fighting and he inadvertently gives Tybalt an advantage so that he mortally wounds Mercutio. 

Tybalt's exaggerated pride, desire for revenge, and hot-headed petulance are directly responsible for Mercutio's death. After Mercutio's death, Romeo confronts Tybalt and, empowered by revenge, kills him in a duel. This incident has a dramatic effect on the plot and drastically changes the lives of our two protagonists. When the Prince learns that Romeo has killed Tybalt, he banishes him from Verona on pain of death if he should return.

It is Romeo's exile that eventually drives both him and Juliet to desperate measures in order to be together. In an ironic and tragic twist, the convoluted plan, suggested by Friar Lawrence, fails miserably and culminates in their deaths. Tybalt's actions, therefore, set in motion all the events which followed. It is in this manner, then, that he is indirectly responsible for the deaths of the two star-crossed lovers.

In Shakespeare's classic play Romeo and Juliet, Tybalt is portrayed as an aggressive hothead who is determined to get revenge on Romeo for sneaking into his uncle's ball. In act 1, scene 4, Romeo expresses his affinity for Juliet aloud, and Tybalt recognizes his voice. Although Tybalt is willing to fight Romeo at the ball, Lord Capulet intervenes by forcing his nephew to control his temper and allows Romeo to enjoy the party. Before Tybalt has the opportunity to challenge Romeo to a duel, Romeo secretly marries Juliet in Friar Laurence's cell, which significantly changes his outlook on the Capulet family.

In act 3, scene 1, Tybalt challenges Romeo to a duel, but Romeo refuses to fight a Capulet because of his recent marriage to Juliet. However, Mercutio defends his friend's honor by accepting the challenge for him and fighting Tybalt. In the middle of their duel, Romeo intervenes, which gives Tybalt the opportunity to fatally wound Mercutio. Following Mercutio's tragic death, Romeo loses control of his emotions and kills Tybalt. Prince Escalus exiles Romeo from Verona, which indirectly leads to his decision to commit suicide. Once Romeo is banished from Verona, he becomes desperate and is no longer in contact with Friar Laurence or Juliet.

Tybalt's decision to challenge Romeo to a duel directly leads to Mercutio's death, which motivates Romeo to take his life. The outcome of Romeo killing Tybalt is his exile. If Romeo were never banished from Verona, he would have known about Friar Laurence and Juliet's plan to ingest the sleeping potion and would have avoided the confusion surrounding her death. Tragically, Romeo never received the letter, assumed that Juliet was actually dead, and committed suicide. When Juliet wakes up in the tomb, she discovers that her husband is dead and also commits suicide. Therefore, Tybalt's challenge was the catalyst that indirectly led to Romeo and Juliet's deaths.

Tybalt is so spoiling for a fight that he can said to be responsible for Romeo and Juliet's deaths. However, I would call him a catalyst rather the responsible party: he hardly forces the two young lovers to kill themselves, which is their own decision. However, he sets in motion a string of events that ended in their deaths.

Tybalt wants to fight Romeo, and when he meets him on the streets of Verona, he tries to bait him into a battle. Romeo, having just secretly married Tybalt's cousin, Juliet, does everything he can to avoid a fight. Mercutio, however, is spoiling for some swordplay, so he decides to engage Tybalt. Romeo interferes to try to stop the fight, but he only distracts Mercutio, which allows Tybalt the chance for the sword thrust that kills him. Romeo feels responsible for his dear friend's death, so he kills Tybalt. As a result, he is banished from Verona.

From that point on, a series of mishaps ensue that lead the lovers to suicide. Is Tybalt responsible? Not really: Juliet could have decided to tell her parents she was secretly married, which would have no doubt made them furious but would have probably saved lives. Further, Romeo, though it would have been completely out of character for him, could have decided to wait and get more information before killing himself, which also would have averted tragedy.

Tybalt is responsible for Romeo and Juliet’s deaths because his insistence on fighting Romeo is what got Romeo banished and led to his suicide.

Tybalt was so caught up in his family feud that he did not think about the consequences. He tried to fight Romeo originally at the party, and then caught up with him again later on the street.  Romeo was not interested in fighting. He also considered Tybalt family.

TYBALT

Romeo, the hate I bear thee can afford
No better term than this,--thou art a villain.

ROMEO

Tybalt, the reason that I have to love thee
Doth much excuse the appertaining rage
To such a greeting: villain am I none;
Therefore farewell; I see thou know'st me not. (Act 3, Scene 1)

Mercutio tried to protect Romeo, and Tybalt ended up fighting him.  He could have just forgotten the whole thing when it was clear that Romeo would not fight.  Mercutio was at the party too.  The idea was to eliminate the Montagues or punish the Montagues.  Tybalt paid with his life for his insistence on maintaining the feud.

Unfortunately, after Tybalt killed Mercutio, Romeo had to step in.  He killed Tybalt, and found himself banished.  It was this banishment that led Juliet to fake her death, and this fake death is that which caused Romeo to commit suicide.  He returned from banishment to find Juliet supposedly lying dead.  Unable to accept that, he killed himself.  Juliet then awoke and killed herself.  Thus, Tybalt was directly or indirectly responsible for his own death and the deaths of Mercutio, Romeo, and Juliet.

In many ways, it could be said that Juliet's hotheaded cousin Tybalt is responsible for Romeo and Juliet's deaths. In Act III, Scene 1, Tybalt challenges Romeo to a duel. Romeo refuses, but Mercutio accepts the challenge in an attempt to protect his friend's honor. Tybalt kills Mercutio, which goads Romeo into combat. Romeo kills Tybalt and is forced to flee the city. Later, we learn Romeo has been banished from Verona and will be killed if he is caught returning. 

Thus, Tybalt is essentially responsible for Romeo's exile, and Romeo's banishment sets up the elaborate death potion hoax that ultimately goes awry and results in the deaths of both Romeo and Juliet. As such, although Tybalt doesn't directly kill Romeo and Juliet, his decision to goad Romeo into a duel is indirectly responsible for the tragedy that unravels in the later acts of the play.