I would call Romeo a static character. He starts off as an impulsive and passionate young man who is in love with love, and ends exactly the same way. Of course, given that the entire time frame of the play is four days, we might not expect a radical change in Romeo in such a short period.
Romeo does change his love interest from Rosaline to Juliet, but his characteristic of falling head over heels in love with a beautiful girl hasn't changed. The undying passion he thought he had for Rosaline simply transfers to someone he finds more dazzling—and who happens to return his love. In both cases, his passion becomes all consuming.
After his secret marriage to Juliet, Romeo does try to show some restraint and not allow himself to be baited into a sword fight with Tybalt. He even tries to break up the fight between Mercutio and Tybalt, but when Tybalt kills Mercutio, the passionate and impulsive side of Romeo once again takes over. Romeo murders Tybalt before thinking through the consequences.
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