Steve is the only character in this story who is not the asshole. "dude" used to be (for a short time, and in certain groups) a unisex term. Its meaning has changed quite a bit over time. You're not the asshole for using the term "dude". Dude's reaction to bring called dude is off the mark, and your reply is at least as far off the mark.
Thank our stars that we have Steve to keep everyone in check.
Counterpoint: maybe he needs to learn that he'd be nothing without the backs of the workers for him to stand on.