"Delta's goal is to always work with customers in an attempt to find solutions to their travel issues. That did not happen in this case and we apologize," said the airline.
It said it had contacted the couple to refund their travel and provide additional compensation.
The apology came the same day members of a U.S. Senate committee held a hearing on the industry's customer service failures and issued harsh criticism of airline fees, disputes with passengers and the lack of competition in the heavily consolidated sector.
The video posted on YouTube by the California couple shows Brian Schear arguing with a police officer and a Delta employee as he sat on Flight 2222 waiting to depart from Maui to Los Angeles.
The dispute was about whether Shear was allowed to use a seat he had originally purchased for his teenage son for his toddler son and whether the toddler needed to be in a car seat or sit on the lap of an adult.
"You will hear them lie to me numerous times to get my son out of the seat. The end result was we were all kicked off the flight," Shear wrote in the caption.
"They oversold the flight. When will this all stop?"
The Schears stayed at a hotel before flying home the next day.