How did the Duke feel about Desdemona and Othello's marriage?

The answer to this question can be found in Act 1 scene 3, which you can read , along with our modern translation and notes on the text, by .

The Duke and Senators are discussing the situation in Cypress when Othello enters with some others, including Brabantio, who mentions that he's angry because Othello has stolen his daughter. Othello explains that Brabantio's daughter, Desdemona, has fallen in love with him and...

The answer to this question can be found in Act 1 scene 3, which you can read , along with our modern translation and notes on the text, by .

The Duke and Senators are discussing the situation in Cypress when Othello enters with some others, including Brabantio, who mentions that he's angry because Othello has stolen his daughter. Othello explains that Brabantio's daughter, Desdemona, has fallen in love with him and they are married. Saying that he is a 'man of action' and not much of a talker, Othello then launches into a very long explanation of how their love came about. At the end, the Duke says that his daughter would fall for Othello, too: "I think this tale would win my daughter too." (line 183) The Duke then tells Brabantio that he should just accept the situation: "Good Brabantio, / Take up this mangled matter at the best:" (lines 184-5)

Brabantio doesn't accept the Duke's advice, and insists on letting his daughter, Desdemona, tell her side of the story. Desdemona states that she chooses Othello and is loyal to him as her husband. The Duke tells Brabantio that he can't change what's happened (the marriage): "To mourn a mischief that is past and gone / Is the next way to draw new mischief on." (lines 219-20) and he must try to be happy: "The robb'd that smiles steals something from the thief; / He robs himself that spends a bootless grief." (lines 223-4)

So overall, the Duke seems satisfied that Othello did not "steal" Brabantio's daughter, but that she willfully entered into marriage with him. The Duke feels that Desdemona and Othello are married, what's done is done, and Brabantio has to accept it and move on.