A protagonist is basically the main character of a story who faces major conflict and who the reader/audience feels empathy for and wants to succeed. An antagonist is another major character who creates conflicts for the protagonist. Traditionally, in William Shakespeare's Romeo and Juliet, most readers would say that Romeo is the play's protagonist because he is the one we want to succeed. We know that his intentions are pure and we feel sorry that he meets his tragic end and does not get what he was fighting for in the first place, which is to be with Juliet. In that same vein, most would label Tybalt the play's clear antagonist, as he is the one who wants to fight Romeo, and the reader is given no reason to ever really feel sorry for, or care about, Tybalt's character. All we ever see from him is his lust for a good fight and his detest for peace.
That said, some could make the argument that Romeo is the one who creates conflict for himself and that his own choices and personality traits bring about his demise. This idea is that of the "tragic hero," and his flaw that destroys him would be his "tragic flaw." If you need to prove that Romeo creates his own conflicts and is the sole character who destroys his chances of being with Juliet, you should first look at his character. Romeo is governed by his emotions at all times though out the story. He shows little ability to think reasonably before acting. We see this immediately, in Act I, Scenes i and ii, where Romeo is so emotional over his unrequited love for Rosaline that he has fallen into a deep depression and describes himself as unable to function:
"Not mad, but bound more than a madman is,/ Shut up in prison, kept without my food,/ Whipped and tormented..." (I.ii.55-57).
When Romeo meets Juliet, he does not stop to think about the dangers of loving someone he should not be with or even of the risk of sneaking back to the Capulet orchard in the middle of the night:
"With love’s light wings did I o'erperch these walls,/ For stony limits cannot hold love out,/ And what love can do, that dares love attempt./ Therefore thy kinsmen are no stop to me." (II.ii.66-69).
Later, he also shows an inability to consider the consequences of his actions when he kills Tybalt after Tybalt stabbed Mercutio. This was completely unnecessary, as the Prince would have probably killed Tybalt anyway since he vowed in Act I, Scene i to kill any Capulet or Montague who fought in Verona's streets again. Of course, he realizes the foolishness of his behavior immediately after, but it is already too late:
"Oh, I am fortune’s fool!" (III.i.98).
This is the decision that dooms him and creates even more conflict for Romeo and Juliet. With Romeo banished and Juliet's family wanting Romeo dead, there is no way they can ever be together, at least not in Verona.
Later, in Act V, when he hears that Juliet is dead, he goes mad and is overcome with emotion. He makes the impulsive decision that he will kill himself, and in doing so he kills Paris as well. If he had only taken a few moments to consider his actions before making such a drastic decision, Juliet may have woken up and the story would have ended much differently.
These are some arguments you could use to describe Romeo as both the story's protagonist and antagonist. He is the one who ultimately creates his own tragedy.