The exact type of cancer the 67-year-old is battling remains unknown.
Band members Mick Fleetwood, Stevie Nicks and Lindsey Buckingham said in a statement Sunday that they hope "fans everywhere will join them in wishing John and his family all the best."
The band last stopped in Los Angeles when they performed at Anaheim's Honda Center and the Hollywood Bowl in May.
The cancelled dates include performances in Australia and New Zealand. The band is also scheduled to perform in Las Vegas on Dec. 30.
The Associated Press contributed to this report.