The letter, issued in conjunction with Icahn's new website "Shareholders Square Table," comes after Icahn urged Cook to use Apple's $150 billion in surplus cash to buy back company shares.
Icahn is scheduled for a live interview on "Fast Money Halftime" with Scott Wapner at noon EDT Thursday.
(Read more: Apple now hordes 10% of US corporate cash)
In the letter, Icahn praises Cook's leadership, but says the discussed share buyback needs to be substantial.
(Read more: Text of Icahn's letter to Apple)
"We want to be very clear that we could not be more supportive of you, the existing management team, the culture at Apple and the innovative spirit it engenders. The criticism we have as shareholders has nothing to do with your management leadership or operational strategy," the letter says. "Our criticism relates to one thing only: the size and timeframe of Apple's buyback program. It is obvious to us that it should be much bigger and immediate."
Apple shares were up Thursday. (For the latest stock price, click here.)
Biggest announced share buybacks in history.
The letter says Icahn's stake in the technology giant is now 4.7 million shares, up from 4 million, making his total stake roughly $2.46 billion. He said he would withhold his shares from the tender offer.
The letter says his increased investment in Apple reflects "our belief the market continues to dramatically undervalue the company."
"From our perspective, Apple is the world's greatest consumer product innovator and has one of the strongest and most respected brand names in history," the letter says.
(Read more: Ex-Apple CEO Sculley: Cook should ignore Icahn)
In an earlier interview with CNBC, Icahn said he would not be dropping this issue.
"I feel very strongly about this," Icahn told CNBC after his meeting with Cook in September. "I can't promise you the stock will go up and I can't promise you they will do the buyback. But I can promise you that I'm not going away until they hear a lot more from me concerning this."
(Read more: Icahn's dinner with Cook)
The letter says Apple's board does not include someone "with a track record as an investment professional."
"In my opinion, any further delay in executing the buyback we hereby propose will reflect this lack of expertise on the board," it adds. "The timing for a larger buyback is still ripe, but the opportunity will not last forever."
Not everyone welcomed Icahn's statement, though. The legendary bond investor Bill Gross took to Icahn's new favorite medium, Twitter, to respond.