A passenger who uploaded video of the aftermath of the incident said a mother had been struck with the pram as it was forcibly removed by the employee.
The video shows the employee saying, "Hit me! Come on, bring it on", when challenged by another male passenger.
American Airlines said it was "deeply sorry for the pain we have caused" the woman passenger and her family.
The incident follows another high-profile clash on a United Airlines plane two weeks ago, when a passenger was violently removed from an overbooked flight.
The American Airlines incident took place at San Francisco International Airport as flight 591 prepared to take off for Dallas/Fort Worth.
Passenger Surain Adyanthaya, who posted the video to Facebook, said that preceding the footage the employee had "violently" taken the pram from the mother, hitting her and narrowly missing her baby.
The footage shows the mother clearly distraught as other employees and passengers try to intervene.
One male passenger demands the name of the employee involved and when a man dressed in American Airlines uniform enters the plane, the passenger tells him: "Hey, bud, you do that to me and I'll knock you flat."
The employee confronts the passenger, telling him to "stay out of this", then saying: "Hit me! Come on, bring it on."
He adds: "You don't even know what the story is."
The passenger replies: "I don't care what the story is. You almost hurt a baby."
In its statement, American Airlines said: "What we see on this video does not reflect our values or how we care for our customers. We are deeply sorry for the pain we have caused this passenger and her family and to any other customers affected by the incident."
It said it was "making sure all of her family's needs are being met while she is in our care".
The airline upgraded the woman and her family to first class when she took another flight to Dallas.
"The actions of our team member captured here do not appear to reflect patience or empathy, two values necessary for customer care. In short, we are disappointed by these actions," the airline said.