We learn in the first act that Desdemona is already missing from her father's house, and Iago tells Brabantio,"...your daughter and the Moor are making the beast with two backs" (1.1 line 114-5). So we can infer that Othello and Desdemona have already consummated their marriage. If they had not done so, you are correct in speculating that the play would have had a different outcome, since Othello would have been able to prove to himself that Desdemona was still a virgin. The issue in the play, however, is not Desdemona's virginity, but her fidelity.
You may wonder why we should believe Iago, given his lies and manipulations, but you will notice that the audience is always aware of these, and thus it is likely to be true that the marriage was consummated.