Woods had been suffering minor discomfort before last week's U.S. Open at Merion, where he aggravated the problem and was seen wincing and shaking his left arm on several occasions.
"I was examined after I returned home from the U.S. Open, and the doctors determined I have a left elbow strain," the American said in a statement on his website.
"I have been advised to take a few weeks off, rest and undergo treatment. I'll be ready to go for the British Open and I'm looking forward to playing at Muirfield."
The 14-times major champion was a big favourite to win the U.S. Open but poor putting and iron play left him 12 shots behind winner Justin Rose.
Woods had been scheduled to play in the June 27-30 AT&T National at Congressional Country Club in Bethesda, Maryland, where he triumphed last year to move past Jack Nicklaus into second place on the all-time PGA Tour winners' list.
The July 18-21 British Open at Muirfield in Scotland is the third of golf's four annual major championships.
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