The 48-year-old is known to have created shows such as "Pushing Daisies", "Hannibal", and "American Gods" and was roped in by NBC in 2015 to set up the reboot as an anthology series.
Apple picked up the series in 2017 in its bid to create its own original content.
According to The Hollywood Reporter, Fuller departed the project over creative differences with Apple, which wanted a more family-friendly take rather than Fuller's darker Black Mirror-style show.
Hart Hanson, who had partnered with Fuller, has also exited the Apple project.
In November last, Fuller along with Michael Green exited "American Gods", reportedly over budget issues.
The original "Amazing Stories", co-created by Steven Spielberg, Joshua Brand and John Falsey, ran for two seasons in the 1980s on NBC, garnering 12 Emmy nominations and winning five.
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