Singer had already filmed a number of scenes, including a recreation of Queen's famed 1985 performance at Live Aid.
Singer denied speculation his exit was down to "a dispute" with lead star Malek - who will play the band's late frontman Mercury in the motion picture - but admitted they did have "creative differences" on set at times.
Singer was sacked Monday for what a source close to 20th Century Fox told ABC News was a pattern of unreliable behavior on the set. Bohemian Rhapsody is scheduled for release in December 2018.
When Ron Howard famously took the reins of Lucasfilm's Han Solo film after original directors Chris Miller and Phil Lord were fired earlier this year, he reshot large chunks of the film under the supervision of the studio - he effectively made it his own.
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"I wanted nothing more than to be able to finish this project and help honor the legacy of Freddie Mercury and Queen, but Fox would not permit me to do so because I needed to temporarily put my health, and the health of my loved ones, first", Singer wrote.
In a statement issued through his lawyer, Singer said he had asked for time off to deal with a "pressing" family matter.
A source also said this week that Tom Hollander, who plays Jim Beach, departed the film due to Singer's behavior, but was persuaded to return.
He added: 'Rami and I successfully put those differences behind us and continued to work on the film together until just prior to Thanksgiving'.
Bohemian Rhapsody stars Malek as Mercury, Ben Hardy as drummer Roger Taylor, Gwilym Lee as lead guitarist Brian May, and Joe Mazzello as bass guitarist John Deacon. Lucy Boynton is playing Mary Austin, lifelong companion of Mercury.