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, a comedy, is very similar to Romeo and Juliet, a tragedy. Much Ado has a main plot set around a set of lovers, Benedick and Beatrice and and a sub-plot set around another set of lovers, Claudio and Hero. Whereas the main plot is pure comedy, the subplot in Much Ado could very well end up in tragedy, as evidenced in Act IV.i.
Act IV in Romeo and Juliet and Much Ado are nearly identical: they involve a staged funeral. Whereas Juliet is faking her death at the bequest of Friar Lawrence, Hero is faking her death at the bequest of her father and uncle. In both cases, the potential groom does not know of the staged death. In R&J, Romeo finds out too late and commits suicide before Juliet wakes up (there's your tragedy). In Much Ado, Claudio repents, agrees to marry Hero's "sister," and all is forgiven (there's your comedy). Tragedy ends with funerals, comedy ends with weddings.
The difference, of course, is that more people are aware of Hero's staged death than are unaware. Only Claudio, Benedick, and Don Pedro believe she is dead. The Friar's plan in Much Ado is well-conceived, and it used to trick the men into submission. Tragedy is avoided when Hero's sister is revealed to be really Hero. Claudio is reconciled with Leonato, Hero, and Benedick.