The American was put into a coma on Saturday after he was knocked out in the 10th round of his super welterweight bout in Chicago.
Promoter Lou DiBella confirmed Day had died on Wednesday “surrounded by his family, close friends and members of his boxing team”.
“He was a son, brother, and good friend to many,” said a statement.
“Pat’s kindness, positivity, and generosity of spirit made a lasting impression with everyone he met.”
Day was taken from the ring on a stretcher after the conclusion of the fight on the undercard of Oleksandr Usyk’s heavyweight victory over Chazz Witherspoon in the US.
“Patrick Day didn’t need to box,” the statement from DiBella Entertainment continued. “He came from a good family, he was smart, educated, had good values and had other avenues available to him to earn a living.
“He chose to box, knowing the inherent risks that every fighter faces when he or she walks into a boxing ring. Boxing is what Pat loved to do. It’s how he inspired people and it was something that made him feel alive.”
Fellow American Conwell, 21, wrote an emotional letter to Day following the fight.
“I never meant this to happen to you, all I wanted to do was win,” he said.
“If I could take it all back, I would. No-one deserves this to happen to them.”
Day won 17 of his 22 fights, with four defeats and one draw.
In July, Russia’s Maxim Dadashev and Argentina’s Hugo Santillan both died following injuries sustained during fights.
“It becomes very difficult to explain away or justify the dangers of boxing at a time like this,” the statement continued.
“This is not a time where edicts or pronouncements are appropriate, or the answers are readily available. It is, however, a time for a call to action.
“While we don’t have the answers, we certainly know many of the questions, have the means to answer them, and have the opportunity to respond responsibly and accordingly and make boxing safer for all who participate.
“This is a way we can honour the legacy of Pat Day. Many people live much longer than Patrick’s 27 years, wondering if they made a difference.