I never meant this to happen to you, all I wanted to do was win," wrote Day's American compatriot Conwell, 21.
"If I could take it all back, I would. No-one deserves this to happen to them."
Conwell posted a link to the letter on social media with the hashtag #champpatrickday.
He said this would be the last time he spoke on the matter because it was "a sensitive topic" for Day's family and friends, himself and the sport of boxing.
"I replay the fight over and over in my head thinking what if this never happened and why did it happen to you," said Conwell.
"I can't stop thinking about it myself. I prayed for you so many times and shed so many tears because I couldn't even imagine how my family and friends would feel.
"I see you everywhere you go and all I hear is wonderful things about you.
"I thought about quitting boxing but I know that's not what you would want.
"I know that you were a fighter at heart so I decided not to, and to fight and win a world title because that's what you wanted and that's what I want so I will use you as motivation every day and make sure I leave it all in the ring every time."