L’Wren Scott, 49, was found dead in her New York flat on Monday in an apparent suicide.
Mick Jagger earlier said he failed “to understand how my lover and best friend could end her life in this tragic way”.
In a statement on their website, The Rolling Stones thanked fans for their support.
They said: “The Rolling Stones are deeply sorry and disappointed to announce the postponement of the rest of their 14 On Fire tour of Australia and New Zealand following the death of L’Wren Scott.
“Mick Jagger, Keith Richards, Charlie Watts and Ronnie Wood wish to thank all of their fans for their support at this difficult time and hope that they will fully understand the reason for this announcement.
“The Rolling Stones are planning to reschedule these postponed shows at a later date.”
The Rolling Stones had been due to begin the tour in Perth on Wednesday, having flown in to the Western Australian city on Sunday.
On his website, Mick Jagger, who started dating Scott in 2001, said they had spent “many wonderful years together”.
L’Wren Scott was found by her assistant at 10:00 a.m. local time on Monday.
Ninety minutes earlier, L’Wren Scott had sent her assistant a text message asking her to come to her Manhattan apartment, without specifying the reason why, the Associated Press news agency reported.
Police said there was no sign of foul play and no note has been found.
The Rolling Stones had planned to perform five concerts in Sydney, Melbourne, Macedon and Brisbane before flying to Auckland, New Zealand, for the final show on April 5.
Ticket holders for the affected shows have been advised to keep hold of their tickets and await further information.
The Rolling Stones’ On Fire tour is scheduled to move to Europe in June, with concerts planned for the Netherlands, Germany, Italy and Belgium.
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