Daly initially said he would try to play "through the pain" of degenerative arthritis in his right knee despite being denied the aid of a cart.
However, the 53-year-old has since had a change of heart and has been replaced in the field by American Kevin Streelman.
Daly, who won the Open in 1995, had been allowed to use a vehicle in the US PGA Championship in May under the Americans With Disabilities Act.
The R&A expressed sympathy with Daly over his condition, but said players "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," in a statement on Saturday.
"Just got word the R&A turned down my application to use a cart at the 2019 Open Championship," Daly tweeted after learning the news.
"Quite disappointed they do not see it the same way our PGA of America and PGA Tour sees it. Different continents different laws???
"I believe all who have earned the right to compete should be afforded that right to compete and give it their best shot.
"While I trust the R&A's decision was made with good intentions, 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 the practice is common.
He last played without a cart in the European Tour's Omega European Masters in Switzerland last September.