Answer: Juliet is the character that says "He was not born to shame: upon his brow shame is ashamed to sit".
Explanation: Juliet pronounces these words to the Nurse after she tells Juliet that Tybalt has died and blames Romeo for it. The Nurse uses harsh words against Romeo and expresses that he should be ashamed. Juliet becomes upset immediately and starts to defend Romeo by saying that he was not born to be shameful and that he only deserves honor ("He was not born to shame: upon his brow shame is ashamed to sit").