In a video circulating on social media, Berghuis is seen lashing out at what appears to be a Twente supporter in front of the Ajax team bus before an official intervened.
"I regret my actions, I should not have done this," Berghuis said in a statement. "After every away game, we get a lot of treats thrown at us at the bus while we take time out to sign autographs for fans.
"I'm used to it by now, but people think they can just shout anything. My reaction doesn't solve anything, I get that. It's not good, I have an exemplary role as a player of Ajax."
An Ajax statement read: "Steven reacted in the wrong way. He quickly made this known to John Heitinga and immediately apologized. Upon arrival at the Arena, he had a conversation with Heitinga, Edwin van der Sar and Sven Mislintat.
"They also wanted to hear his side of the story. The conclusion is that his reaction was wrong and that Ajax disapproves. We will resolve it internally with him."
Ajax finished the campaign in third place in the Eredivisie, outside the two Champions League qualification spots for the first time since 2009.
League champions Feyenoord, who finished 13 points ahead of Ajax, have qualified for the group stage, while second-placed PSV Eindhoven will play in the third qualifying round of Europe's elite club competition.
Ajax qualified for the Europa League playoff round.