The world number two suffered the injury in training on 29 July, and missed two warm-up tournaments in Toronto and Cincinnati.
The 28-year-old had hoped to recover before 25 August, when the US Open gets under way at Flushing Meadows.
"I am very sorry to announce I won't be able to play at this year's US Open," he wrote on his Facebook page.
"I am sure you understand that it is a very tough moment for me since it is a tournament I love and where I have great memories from fans, the night matches, so many things.
"Not much more I can do right now, other than accept the situation and, as always in my case, work hard in order to be able to compete at the highest level once I am back."
Russell Fuller, BBC tennis correspondent
"Rafael Nadal first felt pain in his right wrist while practising in Majorca at the end of last month. He's been wearing a light cast for the last couple of weeks, so his decision to miss the year's final Grand Slam comes as no surprise.
"Nadal is now in the familiar position of having to work his way back from injury, and his absence throws the championships even more wide open.
"The world number one Novak Djokovic hasn't settled on the North American hard courts since his summer wedding, and Andy Murray is still far from his best.
"Roger Federer, a Masters champion for the first time in two years, will be licking his lips."
Nadal has not competed since he was beaten by Australian teenager Nick Kyrgios in the fourth round at Wimbledon on 1 July.
He announced late last month that he would not be defending either his Rogers Cup title in Toronto or his Western & Southern Open crown in Cincinnati.
Both tournaments are key hard-court warm-ups for the US Open, at which Nadal hoped to chase a 15th career Grand Slam title.
The Spaniard beat Serbia's Novak Djokovic 6-2 3-6 6-4 6-1 to win last year's title at Flushing Meadows.
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