A spokeswoman says Campbell is not up for "the very long flight that it would require." The 76-year-old is battling Alzheimer's disease.
It would have been Campbell's last international stop on his "Goodbye Tour."
Rogers is continuing the month-long tour, kicking off Aug. 10 in Christchurch, New Zealand. Australian country artists will fill in as special guests. Campbell was also scheduled to do a solo show in Brisbane, Australia.
Campbell received the Recording Academy's Lifetime Achievement Award this year. He is known for dozens of hits such as "Gentle On My Mind," ''Wichita Lineman," and "Rhinestone Cowboy."