Julio Varela said the incident happened last Tuesday after he spent about three hours trying to correct a ticketing error for his daughter before a family trip to Montreal.
According to Varela, he, his wife and their two daughters were attempting to fly to Montreal on July 14 when they discovered one daughter's ticket had her middle and last names mixed up.
He said the error sent the family back and forth between United Airlines and Air Canada counters for hours.
Varela said an Air Canada employee was working to resolve the issue when a United employee joined the conversation.
"She said 'I already told you. I can't do anything. Whether you fly out or not, I don't care,'" Varela said. "That's what triggered me to say, 'you're rude.' And I also said you're rude and you're lazy."
Varela said he then took out his cellphone and asked the employee for her name.
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Both he and the employee recorded the interaction.
In video of the confrontation, the employee can be heard saying: "Maybe we should call ICE on you."
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When Varela responded, "You going to call ICE on me," the employee continued: "Maybe you need to be. Because you don't act like a citizen."
ABC7 News blurred the employee's face in video of the exchange.
Varela said the comment escalated the dispute.
"That's obviously when I lost it even more because I was already angered and bothered angered by the way she didn't care."
He said the remark was particularly upsetting.
"My jaw dropped. Number one. I was shocked and immediately I was angry," Varela said.
Varela, who identified himself as Hispanic and Mexican, said he was especially troubled by the employee's reference to ICE.
"If she is going to have the gall to say 'I'm going to call ICE' on a Hispanic person, Mexican, and to say that in San Francisco..."
He said the comment was especially concerning amid recent public attention surrounding immigration enforcement actions.
In a statement, United Airlines said: "We're looking into the interaction in this video, but don't have anything additional to share at this time."
Varela said he was not surprised by the airline's response.
He said a United representative contacted him to get his account of what happened and that the two sides communicated briefly by email.
However, Varela said the airline later stopped contacting him.
United has not provided additional details about the incident.