Romeo is a lover, not a fighter. For example, he gets upset at Benvolio in Act I when he sees blood on the ground and deduces that Benvolio was involved in another fight. The feud between the Montagues and Capulets creates prejudice between the two families, and since Romeo doesn't like the feuding, he would also hate the coinciding prejudice.
One example of Romeo facing prejudice is with Tybalt Capulet. Romeo offends Tybalt by crashing a Capulet party on the night he meets Juliet. Since there is a standing feud, Romeo and his friends make sure to keep their masks on during the party, but Tybalt overhears Romeo speaking and would have killed him right then if Capulet himself had not intervened. Capulet tells Tybalt to stand down because he does not want his party to be ruined by Tybalt killing a Montague. Tybalt withdraws until another time.
That other time comes after Romeo has married Juliet--Tybalt's cousin. This gives Romeo even more reason to befriend Tybalt in the face of prejudice. Unfortunately, Tybalt is determined to fight Romeo for crashing his family's party. Romeo tries to dissuade Tybalt by saying the following:
"Tybalt, the reason that I have to love thee
Doth much excuse the appertaining rage
To such a greeting. Villain am I none.
Therefore, farewell. I see thou knowest me not" (III.i.57-60).
Romeo does his best to defend himself with words before the argument turns to blows. He also tries to explain to Tybalt that there is no reason for fighting even though he cannot disclose the reasons at that time. Some would argue that Romeo should have just told Tybalt he married Juliet at that time to diffuse the situation; but there's no guarantee that would have worked. In fact, Tybalt probably would have become more upset had he known about the marriage. Either way, Romeo was faced with Tybalt's prejudice against him as a Montague and there was no way out. Romeo does his best to end the fight, but in the end, Romeo kills Tybalt after Mercutio dies defending him.