Stewart apologised to his fans on Instagram, saying he needed more rest as per his doctor's orders.
He is still scheduled to play on 7 and 8 June, with the first cancelled show rescheduled to 10 June, but there is no alternative date for the cancelled 5 June show.
Stewart is set to play Glastonbury in the Legends slot later this month, an appearance he described as an "honour," despite the high costs of bringing his band from America.
Stewart mentioned he would be scaling back world tours but has no plans to retire, expressing interest in a smaller, more intimate "Great American Songbook, Swing Fever tour" in the future.