Bloomberg’s Businessweek published a story which said malicious chips were built into servers that had been manufactured in China.
But in an unprecedented step, Mr Cook told Buzzfeed: "They need to do that right thing and retract it.”
Businessweek has said it stands by its reporting.
"Bloomberg Businessweek's investigation is the result of more than a year of reporting, during which we conducted more than 100 interviews,” the publication said in a statement.
“Seventeen individual sources, including government officials and insiders at the companies, confirmed the manipulation of hardware and other elements of the attacks.
"We also published three companies’ full statements, as well as a statement from China’s Ministry of Foreign Affairs.
"We stand by our story and are confident in our reporting and sources.”
Businessweek’s story, published earlier this month, sent shockwaves around the technology industry, but almost immediately provoked an unequivocal backlash from the companies said to be affected.
What’s left is an intense stand-off between a magazine famed for its thorough fact-checking, and companies that have offered their denials in the firmest of terms.
In Apple’s case, it’s the first time the company has publicly called for the retraction of a story.
"We turned the company upside down,” Mr Cook told Buzzfeed.
“Email searches, data centre records, financial records, shipment records. We really forensically whipped through the company to dig very deep and each time we came back to the same conclusion: This did not happen. There’s no truth to this.”
Apple’s denials are matched by its peers in the industry - such as Amazon - and also by government agencies in the US and globally.
