Beterbiev edged out Bivol on a majority decision to win the undisputed light-heavyweight world championship after 12 rounds of boxing at the highest level.
That result saw Beterbiev unify Bivol's WBA title with the IBF, WBO and WBC championships he already held.
"I feel empty emotionally now," Bivol told Sky Sports afterwards. "I went to the ring only to win this fight but I don't have my belt."
Bivol's promoter Eddie Hearn is convinced his fighter merited that decision.
"Dmitry Bivol should be undisputed champion," Hearn told Sky Sports. "He deserves it. It was a boxing masterclass.
"You don't deserve to work your whole life to get to that moment, perform like that and get it taken away.
"To give Dmitry Bivol four rounds of that 12-round fight - that judge should never work again."
Hearn added: "No disrespect to Beterbiev because he's a tremendous fighter, and I loved that fight, it was like the ultimate boxing fight. I was just fascinated by it.
"They told him, on the broadcast, in the 10th round: 'You have to knock him out to win this fight'.
"So let's see the rematch."
Bivol also called for a rematch. "If I have this chance of course I would like to fight again," he said.
But, defeated for the first time in his professional career, he declined to dwell on the result of Saturday's fight. "I don't think about did I win or did I lose," Bivol said. "I have to do everything perfect.
"From my side I'm trying to do everything perfect, everything amazing. But I didn't.
"I'm trying to forget about it and move forward now."
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