I would argue that Romeo's love is mature based on two reasons:
1. He asks Juliet to marry him. While he proposes quickly in the relationship, it is still a serious commitment that he is very aware of. He did not ask Rosaline to marry him, which signifies how much deeper his love is for Juliet. Divorce was a rarity in Shakespeare's day; marriage was a commitment only broken by death, so the assumption must be if Romeo was aware of how serious it was, then he was willing to be mature and spend the rest of his life with her.
2. He is willing to end his life if he cannot have her. While some may see this as immature, I'd argue that sacrificing his life for her shows great maturity. And he does it throughout the play, not just at the end. He risks death in the balcony scene, as just the presence of a Montague outside their window would be enough to be killed. And yes, upon hearing about Juliet's death, he would rather not go on living than be without her. He ends his life, which in the end proves wrong, but I believe this is the ultimate act of maturity.