Watson said he needed to spend time with his wife and newly adopted son. Johnson is recovering from back problems.
"Bonding with my son and wife is what is all about right now," Watson said via Twitter. "I plan to take at least a month off. Sorry to disappoint fans, but the Players has one of the best fields all year. The tourney is more than fine without me."
Watson played last week in New Orleans, where he was defending champion, and said at the time his post-Masters activities had left him "exhausted."
Johnson has been out of action since hurting his back while lifting a jet ski at his home. He withdrew from the Masters due to the injury.
"After several weeks of rest and rehab on my back, I am excited to finally be feeling better and have been cleared to get back to work," Johnson said. "It's been a much longer road than I expected.
"I can promise you, I won't be doing any more heavy lifting, other than in the gym."