Petty was visibly emotional during the three-quarter time break, spending some time off the field before returning late in Friday's 58-point win by the Demons over the Lions.
The incident sparked a heated discussion between Melbourne football manager Alan Richardson and his Brisbane counterpart Danny Daly, before Zorko was confronted by several Demons at full-time.
Brisbane released a club statement on Saturday morning in which Zorko admitted he had acted inappropriately.
"I have apologised to Harrison for the inappropriate comment I made to him last night," he said.
"I'm remorseful for my actions and take full responsibility for the hurt caused to Harrison, his family and the Melbourne Football Club.
"In reflection, I understand I need to be a better leader, and have spoken with the club who have reinforced this."
It comes after Demons coach Simon Goodwin confirmed the inappropriate sledge about a family member of Petty's would be handed over to the AFL.
"Petts was a bit emotional at three-quarter time. He went away, re-set himself and came back into the game," Goodwin said.
"That's as much detail as we're going to go into."
Zorko is no stranger to post-match incidents with opponents after having two run-ins with Gold Coast midfielder Touk Miller in the past.
Miller snubbed the handshake offer extended by the young midfielder after the first Queensland derby in 2019, before then delivering a verbal barrage at the Sun in the return fixture that year before bizarrely following Miller around the field with his hand extended after a brief post-game handshake.
Lions coach Chris Fagan said Zorko's behaviour may have been sparked by Brisbane's bad night at the office as the Demons all but extinguished their top-four ambitions, but he defended his leader's character.
"He's very passionate about our footy club and he would have been pretty disappointed at what was unfolding and that might have led to some frustration," Fagan said.
"I'd like to have a look at some vision to have a real accurate feel about that because whilst his leadership might not have been up to standard tonight … he's been great for us over the last four or five years.
"So I'm not going to cane him for one performance where he was potentially below par."
The Demons subbed out Luke Jackson with a corked calf late in the win but Goodwin was confident the ruckman wouldn't miss any game time with next weekend's bye before finals.
Brisbane, who are almost certain to finish fifth or even sixth on the ladder, lost Jarrod Berry to an ankle complaint while Cam Rayner was placed on report for a first-quarter sling tackle on Demon Ben Brown.
