A video of the incident, which was recorded by the mother of the family and posted on YouTube, has racked up more than two million views.
While the incident took place on a flight from Maui to Los Angeles on 23 April, it has gained global attention after the video was published on 3 May.
In the video, the father in the family, Brian Schear, is arguing to keep a seat he had bought for an older son who had taken an earlier flight to ensure the toddler would have a seat.
The crew members insisted that the toddler could not use a seat that was booked under another child's name and also refused to allow the family to use a child safety seat.
According to the BBC, these statements were at odds with Delta's published advice, which says that for children under two years "we recommend you purchase a seat on the aircraft and use an approved child safety seat".
The family was eventually threatened with imprisonment if they refused to disembark.
As their seats were immediately filled by four new passengers, Shearer believes that Delta Airways invented an excuse to throw them off a possibly overbooked flight.
This incident comes at a point when US airlines are under scrutiny for poor treatment of passengers after United Airlines employees forcibly removed a passenger and physically harming him in the process after he refused to give up his seat on an overbooked flight.
Yesterday, Delta released a statement apologising for the incident:
We are sorry for the unfortunate experience our customers had with Delta, and we’ve reached out to them to refund their travel and provide additional compensation. 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.