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The 20-time major champion, whose Golden Grand Slam hopes were dashed by Alexander Zverev on Friday, lost 6-4, 6-7 (6/8), 6-3 as Spain’s Carreno Busta won on his sixth match point.
Djokovic and his mixed-doubles partner Nina Stojanovic then withdrew before their bronze-medal match against Australians Ashleigh Barty and John Peers, according to organisers. It was not immediately clear why.
It means that the men’s world number one will leave Tokyo without a medal.
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The match with Carreno Busta was a gripping encounter played over two hours and 47 minutes in the suffocating Tokyo heat, with both men desperate to secure a medal.
Djokovic, who will bid to win the final leg of the calendar Grand Slam at the US Open, which starts next month, saved match point in the second-set tie-break to force a decider, and then four more late in the third set.
But Spaniard Carreno Busta, who also knocked out second seed Daniil Medvedev earlier in the tournament, struck 32 winners in an excellent performance, with Djokovic managing just 18.
The 30-year-old finally wrapped up victory in a lengthy final game when his illustrious opponent put a forehand into the net.
Djokovic brought back memories of his infamous default against Carreno Busta last year at the US Open, when he inadvertently struck a ball at a line judge.
This time he threw his racquet high into the empty stands as he saw a break point come and go in the opening game of the third set, and continued to cut an angry figure, destroying another racquet after a miss at the net.
He was given a warning by the umpire after that second incident, but not following the first.