The I’m Outta Love singer, who was first diagnosed with the disease in 2003, was due to kick off the tour with a performance in London on April 6.
She said: “I feel so awful to be letting down all my amazing fans who were looking forward to It’s A Man’s World Tour. It just breaks my heart to disappoint them.”
A spokesman for the star said she would continue writing and recording her new album and hoped to schedule a new tour as soon as possible.
He added that she aimed “to make a full recovery with the support of her family, friends and everyone around her” and asked for her privacy to be respected “during such a trying time”.
All tickets for the tour can be refunded from the point of purchase.