The band had already called off concerts over the past week in Pittsburgh; in Kettering, Ohio; outside Chicago; and in Atlantic City, N.J.
The band had hoped to reschedule some of the shows, but decided to cancel the final half-dozen shows on its North American tour. That includes the four U.S. concerts and two shows in Orillia, Ontario.
The band has said Rhodes is suffering from exhaustion or a possible viral infection.
According to the band's website, "Nick held out all week, hoping that he would be well enough to perform again for these final few shows, but unfortunately the medical advice he has been given today requires him to continue to rest."