The 69-year-old Vietnamese-American physician lost two front teeth and suffered a broken nose and “significant” concussion in the incident.
David Dao’s daughter, Crystal Dao Pepper, told the news conference in Chicago they are “sickened” by his treatment.
Video of the incident has been watched millions of times online.
Law enforcement officials dragged Dr. David Dao off Sunday evening’s Chicago to Louisville, Kentucky, flight because it was fully booked, and the airline wanted four passengers to make way for staff members.
The man had refused to leave, saying he needed to go home to see his patients.
Lawyer Thomas Demetrio told journalists on April 13: “He [Dr. David Dao] said that he left Vietnam in 1975 when Saigon fell and he was on a boat and he said he was terrified.
“He said that being dragged down the aisle was more horrifying and harrowing than what he experienced when leaving Vietnam.”
Crystal Dao Pepper, who lives in suburban Chicago, said: “What happened to my dad should have never happened to any human being, regardless of the circumstance.
“We were horrified and shocked and sickened to learn what happened to him and to see what happened to him.”
Dr. David Dao’s lawyers have filed an emergency court request for the airline to preserve evidence ahead of a hearing on April 17.
The filing with an Illinois state court demands that United Airlines and the city of Chicago preserve all surveillance videos, cockpit voice recordings, passenger and crew lists related to the flight.
The airline said it would refund the ticket costs of all passengers on April 9 flight.
Dr. David Dao was released on April 12 from a Chicago hospital, his lawyer said, adding that he planned to have reconstructive surgery.
Thomas Demetrio, and a second lawyer for David Dao, said that neither they nor the Dao family had heard from United Airlines yet.
“For a long time airlines, United in particular, have bullied us,” Thomas Demetrio told the press conference.
“We want respect and we want dignity, that’s it! Not a big deal.”
The lawyer said he did not believe Dr. Dao’s race played a factor, despite an email he had received suggesting he was “the modern day, Asian Rosa Parks”, a reference to the civil rights activist who refused to give up her seat on an Alabama bus in 1955.
Thomas Demetrio also said the family had not heard from United yet.
However, United said in a statement that its chief executive, Oscar Munoz, and the company had “called Dr. Dao on numerous occasions to express our heartfelt and deepest apologies”.
The airline did not comment on the potential litigation.
On April 12, United’s CEO Oscar Munoz said he felt “shame and embarrassment” and vowed it would never happen again.
More than 150,000 thousand people have signed online petitions calling for Oscar Munoz to resign, but he has refused to do so.
Oscar Munoz said Dr. David Dao had not been at fault, adding “no one should be treated that way. Period.”
However, Oscar Munoz initially described Dr. Dao as “disruptive and belligerent”.
United Airlines has confirmed it is offering compensation to customers on United Flight 3411 for their flights.
Three aviation security officers involved in removing Dr. David Dao from the plane have been “placed on leave”, says the Chicago Department of Aviation.
The US Department of Transportation is reviewing whether United Airlines complied with rules on booking.
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