Is Romeo responsible for the majority of the deaths in Romeo and Juliet?

No, Romeo killed Tybalt and Paris.

However, Friar Lawrence made up a plan so that Romeo and Juliet could be together but his messenger didn't give Romeo the message in time. So, technically it was his fault/the messenger's fault that Romeo and Juliet both died in the end ...

Also ...

The first character who is to blame for the deaths of Romeo and Juliet is Romeo himself. The first reason Romeo is to blame is that he went uninvited to the Capulet party. If Romeo did not go to the party, Romeo would have never met Juliet, and Tybalt would never have known he was there. The second reason Romeo is to blame for him and Juliet's death is that Romeo decides to avenge Mercutio's death by killing Tybalt. Romeo should have been able to hold his anger against Tybalt and not kill him. Romeo and Juliet could have ran away together; however, Romeo killed Tybalt causing Romeo to get banished from Verona. Lastly, Romeo can be blamed for killing himself at the end of the play. Romeo was not able to get over the fact that his love had died. Romeo should have been able to hold his anger and not kill himself. Romeo should have been able to at least think about his decision longer to hopefully realize it was not the best decision. In conclusion, If Romeo would have put more time into making the decisions that he made the tragedy might not have occurred.

The next character who is most to blame for the tragedy is Friar Lawrence. The first reason why Friar Lawrence should be blamed for the tragedy is the fact that he marries Romeo and Juliet. Knowing the possible consequences, the friar still decides to marry Romeo and Juliet. The friar knows marring quick can have many consequences upon the lovers, and he could get in a lot of trouble if the two families find out he married the two lovers. Because Romeo and Juliet were now married, their love for each other was even stronger giving them more of reason to kill themselves at the end of the play. Another reason Friar Lawrence is to blame is that he tells Juliet to take the sleeping potion. The friar does not think about what could possibly go wrong with giving her. The friar should have put more thought into his plan, so the plan would not fail. The last reason Friar Lawrence is to blame is that he was not able to get a servant to Romeo to tell him that Juliet is not really dead. He could have sent out multiple servants at different times or just went himself to ensure that Romeo would get his message. In conclusion, Friar Lawrence should have made better decisions during the play to stop the tragedy from happening.

The last character to blame for Romeo and Juliet's death is Tybalt. The first reason Tybalt can be blamed is that he participated in the fighting it the beginning of the play. He played a role in the families hatred toward each other. The feud between the two families would not have been so harsh if members from each family would had agreed to quit fighting. The next reason Tybalt can be blamed is that he sees Romeo at the Capulet party and gets very mad. Tybalt threatens to kill him. If Tybalt was able to realize that Romeo was not causing any harm to the Capulets and be able to hold his temper in, Romeo and Juliet might have survived. The last reason Tybalt can be blamed for the tragedy is the fact that he killed Mercutio. Tybalt did not need to kill Mercutio, it was completely unnecessary. Mercutio's death caused Romeo to get banished from Verona not allowing Romeo and Juliet to be together. Finally, Tybalt should have been able to hold his anger and not be so cruel.

Also, the Capulet and Montague feud is to blame because then tybalt wouldn't have hated Romeo and Romeo and Juliet wouldn't have had to do anything in secret.

So, in a way, everyone has played a part in the death of Romeo and Juliet ...

No, not really. Almost every character has contributed with Romeo and Juliet's death. So no one is mostly to blame.