The club described the Peruvian striker's behaviour as "absolutely unacceptable" and said he would face a heavy fine.
"I had a blackout," Guerrero told the club's website after the latest incident in an unhappy season in which he has also suffered a knee injury and battled against a fear of flying.
"I was insulted and I overreacted. I'm really sorry for it.
"I hope I have the chance to meet the fan in person to apologise. I've already apologised to HSV. Naturally, I will accept any fine from the club."
Chairman Bernd Hoffmann said: "Paolo's behaviour was absolutely unacceptable. That sort of thing should not happen.
"Paolo will be punished by us with a big fine."
Kicker magazine said the fine could be between 50,000 and 100,000 euros and would be donated to charity.
It was only Guerrero's second league match since he tore cruciate knee ligaments in September.
In January, he was forced to delay his return from a stay in Peru due a fear of flying, media reports saying that he had made several failed attempts to board an aircraft.
He now has to compete with Ruud van Nistelrooy for a place after the Dutchman was signed in January.
Hamburg are sixth in the Bundesliga with 45 points from 29 games, three points off a Europa League place and eight points behind third spot which would earn a place in the qualifying round of the Champions League.
Team captain David Jarolim denied the supporters had turned against the team.
"Paolo made a big mistake in the heat of the moment. But there is no rift between the team and the fans. We must stay together to reach our goals," he said.
