Klopp was sent to the stands shortly after his side had fallen behind to Gonzalo Higuain's first-half header in the Group F fixture at the Stadio San Paolo following a furious exchange with the fourth official.
But the man who led Dortmund to last season's Champions League final was contrite afterwards.
"I made myself look like a monkey out there and that's not on," he told ZDF television. "I overstepped the mark and it was really pathetic of me.
"I've already apologised to the team, the referee and the fourth official. I got it wrong.
"I have to accept the blame for this red card, it was just stupid of me."
Dortmund's task was made even harder when goalkeeper Roman Weidenfeller was also red-carded on the stroke of half-time for handling outside his area.
Napoli midfielder Lorenzo Insigne doubled the hosts' advantage with a wonderful free-kick after 67 minutes, with Camilo Zuniga's late own goal all the visitors could muster in response despite a stirring rally in the closing seconds.
"Of course we're all a bit down, but there are positives," added Klopp.
"Without the red card, the injury (to Mats Hummels) and my dismissal, we could have got something here.
"This was probably the warning we needed."
It was Dortmund's first defeat of the season after seven straight wins, five in the Bundesliga, the German Supercup and the first round of the DFB-Pokal.