The former Beatle, who turns 72 later this month, cancelled a string of gigs in Japan and South Korea last month.
He has now said that he will be following doctors' orders to recover fully before he takes to the stage again.
In a message posted on his website, he wrote: "I'm sorry but it's going to be a few more weeks before we get rocking in America again.
"I'm feeling great but taking my docs' advice to take it easy for just a few more days."
The Out There world tour will now resume on July 5 in Albany, New York, and the June 14 to 26 shows in Lubbock, Dallas, New Orleans, Atlanta, Jacksonville, Nashville and Louisville have been rescheduled to October.
Sir Paul, who was treated at a Tokyo hospital, has not performed since cancelling his May 17 gig in the Japanese city due to illness. He called off several other dates in Japan and South Korea.
His tour, which kicked off in May last year, features Sir Paul performing songs from his entire career — as a solo artist, member of Wings and as a Beatle.
Last month Sir Paul said he was "extremely moved" by the messages of support he has received after he was struck down by the illness.