Romeo does indeed mature throughout Romeo and Juliet. When we first meet him, Romeo is a callow, shallow seeming youth who thinks he is in love with Rosaline, but knows nothing of real love. In Act 2, scene 3, when Romeo informs Friar Lawrence that he has "forgot" the name of Rosaline and is now in love with Juliet, the Friar replies, "Holy Saint Francis, what a change is here! Is Rosaline, that thou didst love so dear, so soon forsaken? Young men's love then lies not truly in their hearts, but in their eyes. Jesu Maria, what a deal of brine hath wash'd thy sallow cheeks for Rosaline!" Now, unless one chooses to believe that Shakespeare's intent was to have Romeo and Juliet's love for one another be but another example of teenage infatuation, which would hardly seem the case given that this is the most popular love story of all time and that they are both willing to die rather than live without the other; one can only conclude that Shakespeare's purpose in starting with Romeo's infatuation for Rosaline was to contrast the idea of love with true love.
However, the scene where Romeo's newfound maturity shines most clearly is in his confrontation with Paris outside of the Capulet tomb in V.iii. When Paris attempts to arrest Romeo, who has illegally returned to Verona after being exiled, he says: "Condemned villain, I do apprehend thee. Obey and go with me, for thou must die." Romeo replies, "I must indeed, and therefore came I hither. Good gentle youth, tempt not a desp'rate man. Fly hence and leave me. . .I beseech thee, youth, put not another sin upon my head by urging me to fury: O, be gone!...Stay not, be gone; live, and hereafter say a madman's mercy bid thee run away." Here, Romeo refers to Paris as a "youth" twice and to himself as a "man" twice ("desp'rate man" and "a madman's mercy"). Clearly, Shakespeare wants us to see Romeo as the man in the tragic situation. When we contrast his words and actions in the tragic conclusion to the play with the shallow figure he presented initially, it seems unquestionable that his love for Juliet has matured him dramatically.
In my opinion, no. Other readers may disagree with me. That's the fun part about literature; although Beatty from Fahrenheit 451 would definitely disagree. I'm not a fan of Romeo, because I think that he never matures past the Petrarchan lover that Shakespeare introduces him as.
At the start of the play, Romeo is introduced as a love sick, whiney teenager. He is "desperately" in love with Rosaline to the point that he is inconsolable. He's acting like a huge drama queen. Even Friar Laurence knows this about Romeo. Laurence even tells Romeo that Romeo says lovers' quotes to Rosaline without really knowing what they mean.
To me, Romeo moves on from Rosaline to Juliet in the blink of an eye. She is just another beautiful young lady for him to emotionally throw himself at. Romeo says the same kind of romantic stuff to Juliet that he said to Rosaline.
I see Romeo as an emotionally driven character who makes snap judgments. Throughout the play, I just don't see how that changes. When Romeo learns of Juliet's death, he goes to an apothecary for poison. He DOES NOT do what he has done throughout the play, which is seek advice from Friar Laurence. Had he done that, Romeo would have learned Juliet's plan and actual fate. Instead, though, Romeo acts the same way he did with Rosaline's denial of him. Romeo becomes so depressed and lovesick that he sees no reason for living anymore. The answer? Suicide. Over a girl he has known for three days.
No, I don't think Romeo matures throughout the play.