Is one's 'Umrah considered valid if they cursed someone with the words "son of a dog" while in a state of ihram?

Your brother's 'Umrah is valid, and he is not required to repeat it. Disobeying Allah does not affect the validity of the ritual, even if it diminishes the reward. Your brother must repent to Allah for the sin. His 'Umrah is valid by the consensus of scholars. Ibn Hazm dissented on this point, but Shaykh al-Islam refuted him, stating that the consensus on its validity is more apparent.