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.)