In the letter, seen by Reuters, Mr Munoz did not apologise for the way the passenger was handled, instead claiming he had "defied" security officers.
The airline's chief executive said there were lessons the company could learn, though he stressed that he "emphatically" stood behind his employees.
Shares in United Continental (NYSE: UAL - news) had initially held firm after the incident in which the traveller was forcibly removed from the overbooked US domestic flight after refusing to give up his seat.
But investors became more anxious as people kept watching and sharing the footage - including 130 million views on China's Weibo social media platform by Tuesday afternoon.
The company earned about 14% of its annual revenue last year by flying Pacific routes.
In his letter to staff, Mr Munoz wrote: "We sought volunteers and then followed our involuntary denial of boarding process (including offering up to $1,000 in compensation).
"When we approached one of these passengers to explain apologetically that he was being denied boarding, he raised his voice and refused to comply with crew member instructions."
The Chicago Department of Aviation said one of its officers had not followed protocol and had been placed on leave pending a review of his actions.
Flight 3411 from Chicago to Louisville was preparing for take-off at O'Hare International Airport when cabin crew informed the passengers that the flight was overbooked.
The crew asked four passengers to free up space for staff who needed to be at work the following day.
As no-one offered to give up their seat, the airline choose four passengers at random.
The man who was selected refused to leave, prompting airport security guards to remove him from the plane.
A video of the incident shows the man screaming as his seatbelt is unbuckled and he is pulled from his seat by three guards.
He appears to hit his face on the adjacent seat, before he is seen being dragged down the aisle by his arms.
Witness Jayse Anspach said the passenger was a doctor who said he had to be at work the following day.
Mr Anspach added: "The doctor's face was slammed against an armrest, causing serious bleeding from his mouth.
"It looked like he was knocked out, because he went limp and quiet and they dragged him out of the place like a rag doll."
Mr Anspach said the passenger ran back into the plane 10 minutes later "with a bloody face".
Another witness, Audra Bridges, said the incident left passengers "shaky and so disgusted".
In an official statement United Airlines chief executive Mr Munoz said: "This is an upsetting event to all of us here at United.
"I apologise for having to re-accommodate these customers. Our team is moving with a sense of urgency to work with the authorities and conduct our own detailed review of what happened.
"We are also reaching out to this passenger to talk directly to him and further address and resolve this situation."
It is the second time in recent weeks that United Airlines has been engulfed in controversy.
Last month, the airline was accused of sexism after barring two teenage girls from boarding a flight because they were wearing leggings.
The incident sparked a social media storm. The airline said the girls were travelling under an employee travel pass that includes a dress code.