Daly said he might soon need a total knee replacement but will attempt to play in the 148th Open July 18-21 at Royal Portrush in Northern Ireland.
"My plan is to give it a shot in 2 weeks time at Portrush," Daly tweeted. "Fingers crossed I can make it thru the pain."
The 53-year-old American, who won the 1995 British Open, was approved to use a cart at the PGA Championship in May at Bethpage Black, the PGA of America complying with the Americans with Disabilities Act. Daly missed the cut.
"Just got word the R&A turned down my application to use a cart at the 2019 Open Championship," Daly tweeted. "Quite disappointed they do not see it the same way our PGA of America and PGA Tour sees it. Different continent different laws???
"I believe all who have earned the right to compete should be afford that right to compete and give it their best shot. While I trust the R&A's decision was made with good intensions, I could not disagree more with their conclusions."
Daly, whose other major title came at the 1991 PGA Championship, regularly uses a cart on the PGA Champions tour, a 50-and-over circuit where carts are common.
Daly last played without a cart in the European Tour's Omega European Masters last September in Switzerland.
Jim Bradley, orthopedic surgeon for the NFL Pittsburgh Steelers, has reconfirmed the medial and patella femoral degenerative arthritis in his right knee, Daly said in his posting.
"The R&A appreciate the difficulty John is facing and have full sympathy for him as this is clearly a serious, long-term condition," the golf governing body said in a statement saturday.
"Having considered all of the relevant factors, the Championship Committee has decided to decline his request.
"The R&A believe that walking the course is an integral part of the Championship and is central to the tradition of links golf which is synonymous with The Open.
"We must also ensure that, as far as possible, the challenge is the same for all players in the field."
The R&A noted the terrain at Royal Portrush is not suited to carts, saying some area are inaccessible, and the club doesn't allow them.
"This is not a decision we have taken lightly but we believe it is the right one for the Open," the R&A said. "John has a special place in our hearts as a champion Golfer and he will always be welcome at the championship both at Royal Portrush and in future."