The airline's chief executive, Oscar Munoz, is insisting he will not resign.
As of Tuesday, Dr Dao was still recovering in a Chicago hospital, his lawyer said, but a family member is expected to give a news conference on Thursday.
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 City of Chicago runs O'Hare International Airport.
The airline has confirmed it is reaching out to customers on United Flight 3411 and offering compensation for their flights.
On Wednesday, Mr Munoz said he felt shame and embarrassment and vowed it would never happen again.
Mr Munoz had initially described Dr Dao as disruptive and belligerent.
Two online petitions calling for Mr Munoz to resign have gained 64,000 signatures and 75,000 signatures respectively.
But when asked if he would stand down, he said: I was hired to make United better and we've been doing that and that's what I'll continue to do.
Dr Dao was pulled off Sunday evening's flight because it was fully booked, and the airline wanted to get four passengers to leave to make room for staff members.
Dr Dao was left bloodied after law enforcement officials dragged him off the plane as he refused to leave the flight bound for Louisville, Kentucky.
The footage provoked international outrage and the Dao family issued a statement expressing gratitude for the outpouring of support.
Asked what the company would do in future if a seated passenger refused voluntarily to leave an overbooked plane, Mr Munoz said: We're not going to put a law enforcement official to take them off.
He added that Dr Dao had not been at fault: He was a paying passenger sitting on our seat in our aircraft and no one should be treated that way. Period.
But on Monday Mr Munoz said Dr Dao's conduct meant employees had been left with no choice but to call Chicago Aviation Security Officers to assist in removing the customer from the flight.
On Wednesday, two of the aviation security officers involved in removing Dr Dao from the plane were placed on leave, the Chicago Department of Aviation said, adding to one other that was suspended on Monday.
The US Department of Transportation is reviewing whether United complied with rules on overbooking.
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