Act IV, scene i of Much Ado, though it is a scene that is tragic in tone, does not belong, necessarily, in a Tragedy any more than the scene in Hamlet with the Gravediggers (clowns) or the knocking scene in Macbeth with the drunk, comic Porter belong in Comedies. Shakespeare may have more commonly added comic relief to his Tragedies, but he was interested in playing against the expected tone in all of his plays, and this means that you will find tragic moments in his Comedies as well as comic moments in his Tragedies. The wedding scene in Much Ado is one of the most blatantly tragic scenes in any of his comedies.
Up to this point, the play has focused on high-jinks. The only simmering darkness has been the mischief stirred up by Don John. However, when Don John tries to create conflict by convincing Claudio that Don Pedro is trying to win Hero's hand for himself, this plot is quickly and easily foiled. Don John, from this, appears to be an ineffectual villain, and, as such comic. So, even the potential of Don John to stir up trouble doesn't seem very possible early in the play.
But Shakespeare turns all of this around with a scene that is not actually shown in the play, an encounter that Don John swears to Claudio and Don Pedro that will prove that Hero is unfaithful. We hear about the faked encounter from Borachio (Don John's henchman) when he is taken into custody by Dogberry and his men.
And so, by Act IV, the audience is waiting to see what Claudio will do at the wedding. He has sworn to disgrace her if she is proven to be false, and Borachio has related that he saw Claudio say he "would meet her as he was appointed next morning at the temple, and there, before the whole congregation, shame her with what he saw." And yet, Borachio is a villain's henchman. Can we, the audience, believe that Claudio will actually do this?
So, when Claudio rejects Hero in so vile a fashion, we are, as you point out, in the realm of the tragic not comic. And yet, Shakespeare has been very cunning, to introduce this dark moment in a play of such, otherwise, lighthearted looking at love. Claudio's suspicious and jealous nature does reflect the darker side of love, and Shakespeare is not afraid to interject some of this dark side of love into the play.
But it definitely does not make the play a Tragedy in its form. A Comedy, by definition ends in the pleasant resolution of misunderstandings and at least one wedding. Much Ado fits this structure, even providing two weddings at the end -- that of Beatrice and Benedick, and finally, of Claudiuo and the "reborn" Hero.
Much Ado about Nothing Act 4, Scene 1: "Within a Church"
Prince Don Pedro, John the Bastard, Leonato, Friar Francis, Claudio, Benedick, Hero, and Beatrice enter the church for the wedding of Hero and Claudio. As it begins, the friar asks if anyone present wishes to bring reason as to why the two should not be married. Claudio steps forward to everyone's shock and confusion. He pushes Hero away, disgracing her and her family. "There, Leonato, take her back again: / Give not this rotten orange to your friend; / She's but the sign and semblance of her honor" Act 4, Scene 1, lines 31-33. Everyone is confused and hurt because of Claudio's rage except Don Pedro and John the Bastard. Don Pedro defends Claudio because he, too, saw "Hero" with another man at her window the night before. John revels in his villainy. Claudio, Don Pedro, and John leave the church in anger. Hero faints and Beatrice rushes to her side. Leonato is so furious with his daughter's supposed actions that he wants her to die. Her shame is too great for him to handle. Benedick and Beatrice try to help Hero by defending her innocence. Benedick asks Beatrice if she was with Hero last night. She responds that she has been her roommate every night for the past year except for last night. Leonato sees this information as proof of Hero's infidelity and again wants her to die. The friar comes up with a plan, for he too believes Hero to be innocent.
Topic Tracking: Faithfulness 8
"Your daughter here the princes left for dead;
Let her awhile be secretly kept in,
And publish it that she is dead indeed:
...Change slander to remorse...
She dying, as it must be so maintain'd,
Upon the instant that she was accus'd,
Shall be lamented, pitied and excus'd
Of every hearer" Act 4, Scene 1, lines 204-219
Benedick persuades Leonato to go along with the plan of feigning Hero's death and funeral to save her name and bring forth the truth. The friar, Hero, and Leonato leave the church with possible hopes of a future second wedding.
Topic Tracking: Mistaken Identity 7
Beatrice and Benedick are left alone in the church to sob for Hero. Beatrice is hurt and angry for her cousin, Hero, while Benedick simply wants to help. They fumble with their words until Benedick reveals his love for Beatrice. She returns the affection. "I love you with so much of my heart that none is left to protest" Act 4, Scene 1, lines 291-292. Beatrice tells him to kill Claudio to avenge Hero. He will not do so. Beatrice says that if he loves her, he will do this for her. She erupts into a tirade of words and bitterness for her wronged cousin until Benedick says that he will challenge Claudio. The two are engaged.
Topic Tracking: Battle of the Sexes 7