Romeo refuses to fight Tybalt because he does not want Juliet to be angry with him as Tybalt is now Romeo's cousin -in- law.
Romeo refuses to return Tybalt's verbal abuse because he does not want to start a fight because he knew the prince had already warned them not to fight and he was his cousin and he knew something bad was going to happen.
Also, Romeo does not want to start fight with Tybalt because he is now married to Juliet, Tybalt's cousin, and he wishes to maintain peace with his new family member, even though Tybalt is unaware that they are now related by marriage. When Tybalt murders Romeo's best friend, Mercutio, he quickly changes his mind.
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