Woods, who had surgery in March to treat a pinched nerve in his back, asked Watson not to consider him for the biennial team competition against Europe next month.
"While I greatly appreciate Tom thinking about me for a possible captain's pick, I must take myself out of consideration," Woods said in a statement.
"I've been told by my doctors and trainer that my back muscles need to be rehabilitated and healed. They've advised me not to play or practise now. I'm extremely disappointed that I won't be ready for the competition."
Woods, a 14-times major winner, has been a shadow of his former self since his back surgery in late March and missed the cut at last week's PGA Championship, only the fourth missed cut of his professional career in a major.
Though he had pronounced himself fit to compete in the year's final major after recovering from back spasms that forced him to withdraw from the WGC-Bridgestone Invitational the previous week, he played poorly in both rounds.
By some distance the leading player of his generation and arguably of all time, Woods had previously missed the cut as a professional at the 2006 U.S. Open, 2009 British Open and 2011 PGA Championship.
Woods, 38, has appeared on seven U.S. Ryder Cup teams between 1997 and 2012 but missed the 2008 edition at Valhalla Golf Club after having knee surgery earlier that year.
The Ryder Cup will take place at Gleneagles in Scotland from Sept 26-28.