Day earned a captain's selection by Els but will be unable to compete in his home country when the International Team hosts the Americans at Royal Melbourne in two weeks.
"We wish Jason well and hope his back recovers quickly. We were eager to have Jason as part of the team at Royal Melbourne and his experience will be missed," Els said.
Day will also miss the Australian Open, the nation's biggest golf tournament, which begins Dec. 5.
In Day's absence, South Korea's An will make his Presidents Cup debut. Els added the 28-year-old to a roster that already includes six rookies. An finished 15th in the points standings; the top eight earned automatic spots on the team.
The Presidents Cup gets underway Dec. 12 in Australia.