In this passage, Leonato still believes Claudio's and Don Pedro's accusations and is expressing his grief at having had a child who could grow up to allow herself to be defiled and ruin even his own reputation. Shakespeare particularly shows Leonato's beliefs and dispair in this passage using repetition of rhetorical questions and other rhetorical schemes involving repetition as well.
We first see the repetition of rhetorical questions in the first few lines. When friar demands of Leonato why Hero should not open her eyes after having fainted, Leonato rhetorically asks:
Wherefore? Why, doth not every earthly thing
Cry shame upon her? Could she here deny
The story that is printed in her blood? (IV.i.126-128)
In this passage, since Leonato believes he already knows the answers to his own questions, Leonato is rhetorically asking why every living thing should not now be shaming Hero and if she dared deny her accusations. Rhetorical questions like this are used all throughout this speech, showing us the nature of Leonato's feelings at this moment.
Leonato also repeats the word "mine" in this passage, showing us that he is concerned about what Hero's accused reputation will do to his own. Shakespeare uses a rhetorical scheme referred to as anaphora in which beginning clauses or phrases are repeated, in order to emphasize Leonarto's present focus on himself and his own reputation. We see anaphora being used in the lines:
But mine, and mine I loved, and mine I praised,
And mine that I was proud of--mine so much
That I myself was to myself not mine. (142-144)
We see anaphora being used with the repetition of the word "mine" in the beginning of each clause. The repetition of the word "mine" shows us that he is feeling betrayed by his own daughter and feeling grieved over how her "fallen" actions will destroy his own reputation, especially as Governor of Messina.