In response to the letter, Apple wrote to CNBC that it "always appreciate[s] hearing from our shareholders. Since 2013 we've been aggressively executing the largest capital return program in corporate history. As we've said before, we will review the program annually and take into account the input from all of our shareholders."
Icahn, who owns about 53 million shares in the company, has previously pushed Apple on its stock buybacks.
"You have said before that the company likes to be 'opportunistic' when repurchasing shares and we appreciate that. With this letter we simply hope to express to you that now is a very opportunistic time to do so. We think a tender offer is simply a good method of conducting a large repurchase in an expedited timeframe, but the exact method and the exact size is not the key issue for us. We are simply asking you to help us convince the board to repurchase a lot more, and sooner," the letter said.
In order to "preemptively diffuse any cynical criticism" about the request, the activist investor committed not to tender any of his own company's shares if Apple takes his recommendation.
Icahn explained that the letter is not intended to criticize the direction of the company, and in fact represents a strong belief in its future. Apple, Icahn estimated in the letter, will see sales grow 82 percent over the next three years—the company's sales have already increased 65 percent in the last three years.
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"We feel compelled to do so because we forecast such impressive earnings growth over the next few years, and therefore we believe Apple is dramatically undervalued in today's market, and the more shares repurchased now, the more each remaining shareholder will benefit from that earnings growth," the letter said.
CNBC's Jim Cramer said Icahn's note "is not scathing, but it basically says, 'Alright, [Apple CEO Tim Cook]. I've been around forever. You're a newcomer. Do it my way or the highway.'"
The letter also goes into detail concerning what Icahn thinks about several major Apple products. On the just-announced Apple Watch, the activist investor predicted another Cupertino-led revolution.
"It appears to us that Jony Ive and his team have yet again executed at a level that will bring to market a product that revolutionizes the entire category from both a hardware and software perspective, especially given that Apple has developed an entirely new operating system for this device, and catered to the notion that such a device needs to be far more fashionable and personal than other products currently available in this category," the letter said.
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Icahn predicted that Apple will sell about 20 million watches in fiscal 2015, and then 45 million the following year.
As for the continual rumors that Apple may be getting into the television set business, Icahn wrote that "we have good enough reason to expect the introduction of an UltraHD TV set" in fiscal 2016.
On Wednesday, Icahn announced on Twitter that he would be sending an open letter to Apple CEO Tim Cook.