Musk's appointment was set to become official on Saturday, but he told Twitter that morning he would not join, Twitter CEO Parag Agrawal tweeted late on Sunday.
"There will be distractions ahead," he continued, but urged staff to "tune out the noise."
He's criticized the way the company enforces its rules about what people cannot say on the platform, suggesting it has failed to "adhere to free speech principles," and has asked whether it should make its algorithm open source.