Final answer:
Brabantio does not arrest Othello because the Duke needs Othello for state matters, and Desdemona testifies she married Othello willingly.
Step-by-step explanation:
When Brabantio confronts Othello in the streets, he does not arrest him as he originally intended because the Duke has sent for Othello on matters concerning the state. Brabantio initially accuses Othello of bewitching his daughter, Desdemona, but is persuaded to bring the matter before the Duke. Once they all meet, Desdemona herself testifies that she willingly married Othello, which mitigates Brabantio's claims and prevents him from taking any legal action against Othello at that point.