Russia's second-ranked Medvedev dominated Djokovic 6-4, 6-4, 6-4 to prevent the first calendar-year Grand Slam since 1969 and keep Djokovic level with Roger Federer and Rafael Nadal atop the career Slam title list.
"Sorry for you fans and Novak because we all know what he was going for," Medvedev said.
"What you have accomplished in your career... for me, you are the greatest tennis player in history."
Medvedev, the 2019 US Open runner-up, captured his first Grand Slam title in his third Slam final, a rematch of February's Australian Open final, won in straight sets by the 34-year-old Serbian to launch his Slam quest.
"Congratulations to Daniil. Amazing match," Djokovic said. "If there's anyone that deserves a Grand Slam title right now, it's you."
Djokovic's bid to sweep the US, French and Australian Opens, and Wimbledon in the same year for the first time since Rod Laver did it 52 years ago collapsed at the last hurdle, with 83-year-old Aussie legend Laver among 23,700 watching at Arthur Ashe Stadium.
Djokovic missed out on his fourth US Open title and will enter 2022 tied with "Big Three" rivals Roger Federer and Rafael Nadal, both absent with injuries, on an all-time record 20 men's Slam titles.
A tearful Djokovic paid tribute to the crowd that roared support for him even when all seemed lost down two sets and two breaks.
"Even though I have not won the match, my heart is filled with joy and I'm the happiest man alive because you guys have made me feel special on the court," Djokovic said.
"You guys touch my soul. I've never felt like this in New York."
Instead of joining immortals Laver and Don Budge, who completed the first men's Slam in 1938, Djokovic became the third man to fall at the final match attempting to complete the Slam after Australians Jack Crawford in 1933 and Lew Hoad in 1956.
Djokovic saw his 27-match Slam win streak halted three shy of the record he set in 2015 and 2016 when he won four consecutive Slams, ending with the 2016 French Open.
Medvedev, 25, became the third Russian man to win a Grand Slam title after Yevgeny Kafelnikov at the 1996 French Open and 1999 Australian Open and Marat Safin at the 2000 US Open and 2005 Australian Open.