Moments after The Unit made his big debut on the new season of "Jersey Shore" ... dude hit the streets of the L.A. last night ... and immediately took his first public shot at Snooki.

The Situation's pal was leaving STK with porn star Amia Miley -- when the paps asked about his big fight with the Snookster on last night's show (below).

Unit quipped, "The thing with me and Snooki ... she can't seem to take my name out of her mouth ... but that's Snooki for you, she's always got something in her mouth."

And if you thought that was classy ... dude also compares himself to Babe Ruth and says strippers PAY HIM when he goes to the nudie bar.