Did this character experience any growth or changes during the story (explain)?

No, Leonato is a loving father and gracious man throughout the play. The only time he disappoints the reader (or audience) would be his momentary accusation against Hero and her innocence. At that moment he becomes a prideful man rather than a loving father (I bet he'll never jump to conclusions again). His belief that Hero is dead is devastating to him. None-the-less, his character is a minor one, and he doesn't grow or change during the course of the play.