Romeo is shamed in Act III, Scene I, because Tybalt had challenged Romeo to a duel in a letter that Romeo never received.  Therefore Mercutio decides to take over the duel with Tybalt and ends up being stabbed under Romeo's arm when he goes between them to break up the fight.  Romeo is trying to keep the peace because he has just married Juliet and is now related to Tybalt.

Here is the quote relating to Romeo's shame:

"O sweet Juliet,/ Thy beauty hath made me effeminate/And in my temper softened valour's steel."

Readers could also say that Romeo is not only shamed by his "effeminate" behavior, but also by giving into his passion. In one of his most famous quotes, Romeo announces:

"O, I am fortune's fool!"

This is after he has killed Tybalt. This exclamation demonstrates Romeo's acknowledgment that he has behaved without thinking, allowing others and situations to dictate his actions, and thus leading to his own downfall.

In Act III when Romeo tries to avoid a fight with Tybalt, Mercutio takes up the challenge, at which time, Romeo comes between them and Mercutio is stabbed under his arm. Later, when Romeo is told that Mercutio is dead, he realizes that his love for Juliet has made him act “effeminately.” This causes him to feel shame and leads him to take steps toward a deadly twist of fate that will lead to his downfall in the end. When Mercutio is killed, Romeo’s sense of honor and loyalty leave him no choice but to avenge his friend’s death. He calls to Tybalt who returns and they fight. Romeo kills Tybalt and immediately realizes that he has murdered his new bride’s cousin.