If he prays without wudoo’ by mistake, he has to repeat his prayer
Yes, you have to do wudoo’ and repeat your prayer. The scholars are unanimously agreed on that, because purity (tahaarah) is one of the conditions of prayer being valid. The evidence for that is the words of the Prophet (peace and blessings of Allaah be upon him): “Allaah does not accept the prayer of any of you if he breaks his wudoo’, unless he does wudoo’ again.” Narrated by al-Bukhaari, 6954; Muslim, 225.
Muslim (224) narrated that ‘Abd-Allaah ibn ‘Umar said: I heard the Messenger of Allaah (peace and blessings of Allaah be upon him) say: “No prayer is accepted without purification.”
Al-Nawawi (may Allaah have mercy on him) said in al-Majmoo’, 2/79:
The Muslims are unanimously agreed that it is haraam for a person who has broken his wudoo’ to pray, and they are unanimously agreed that his prayer is not valid, whether he is aware that he broke his wudoo’ or is unaware of that or forgot. But if he prayed because he was unaware of it or forgot, then there is no sin on him, but if he was aware that he had broken his wudoo’ and that it was haraam for him to pray in this situation, then he has committed a grave sin.
And Allaah knows best.