Garfield is a bully; all trickster characters who achieve special mastery over their universe (Bugs Bunny, Axel Foley, Brer Rabbit, etc.) press their intellectual advantages to some degree, and could technically be charged with emotional and mental bullying. But Garfield regularly physically assaults Jon, Odie, Nermal, spiders, mailmen and others when manipulation fails or is just too taxing for his liking. This is one of the qualities -- if not the quality -- that makes Garfield a uniquely and specifically American pop culture icon, and is the source of a lot of his power as an instrument of social criticism.
On one hand, we have the small, bitter ironies of a self-fulfilling prophecy: both the dog's sign (which in part inspires the cat to retaliate), and Garfield's threat (a cruel response to accusations of cruelty) elicit exactly the reply they were intended to avoid. The supreme comic contradiction of Garfield is his utter narcissism despite failing to manifest many positive characteristics. The masterstroke, as in Confederacy of Dunces, is to position the ill-tempered slob as the hero, by placing him in a world so screwed-up that his stubborn egoism looks like integrity. Whether that little gay dog takes the sign down or not (and who made that sign for him?), Garfield, ever a credit to his race, is going to pound the bejeezus out of him anyway.