"I said, 'So what? The board is not God,'" Icahn said during an interview on the financial news channel CNBC. "The board should be listening to what their shareholders want."
Icahn said Apple agreed to get back in touch with him in three weeks.
Apple spokesman Steve Dowling declined to comment.
"I feel very strongly about this," Icahn told CNBC. "I can't promise you the stock will go up and I can't promise you they are going to do the buyback, but I can promise you that I am not going away until they hear a lot more from me concerning this."
Icahn, 77, didn't say what he will do if Apple rejects his ideas. In past investments, Icahn has spearheaded shareholder rebellions aimed at ousting corporate boards who spurn him. Icahn told CNBC that he believes Apple's board is smart and praised Cook for the job he has done since he succeeded company co-founder Steve Jobs as CEO two years ago.
Icahn believes the company's stock will eventually soar again and cited consumer loyalty to Apple's brand. He considers Apple's shares currently as cheap as Netflix Inc.'s stock when he acquired a nearly 10 percent stake in the Internet video subscription last year for an average price of about $58 per share. Netflix's stock is now trading above $320, reaping Icahn a gain of about $1.5 billion.
Investors are evidently hoping that Icahn's crusade will prod Apple to do something to lift its stock. The company's shares climbed $11.21, or 2.4 percent, to close at $487.96 Tuesday. The stock has gained 4 percent since Icahn got involved.
Icahn told CNBC Tuesday that his fund has invested about $2 billion in Apple so far, but he didn't reveal how many shares he owns. A $2 billion investment in Apple currently represents about 0.5 percent of the company's market value of about $444 billion.
The public critiques by Icahn threaten to compound Apple's headaches. The company is trying to regain some of the market share that it has lost in the mobile computing market during the past year to less expensive smartphones and tablet computers running Google Inc.'s free Android software.
The tougher competition has slowed Apple's growth and decreased the average prices for its iPhones and iPads, trimming the company's profit margins.
Apple is hoping to bounce back with the recent release of two more iPhones, a high-end model called the 5S and slightly less expensive device called the 5C. Apple said it sold more than 9 million of the new iPhones during their first weekend on the market last month, the fastest start yet for its marquee product line.
Despite the downturn in its stock price, Apple remains the world's most valuable company. Apple also has about $147 billion in cash, but the company isn't relying on much of that money to buy back its stock. Instead, the company issued bonds to take advantage of low interest rates, a borrowing tactic that Icahn believes Apple should use again to raise even more money for stock buybacks.