After previously calling off the final four dates of their US leg, their two Canadian dates scheduled for Friday and Saturday will also not go ahead.
The band said doctors had told Rhodes he needed to continue to rest for at least another 72 hours.
They added that they were "devastated" their two-year tour had ended this way.
Doctors had previously told Rhodes the illness could be the result of a viral infection.
A statement on the band's website said: "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.
"The band are devastated that what has been a triumphant tour for them, with many highlights, has ended this way - and know that the fans will be as disappointed as they are."
Earlier in the week, Rhodes said he felt "completely helpless over the situation and all I can do is apologise to our fans".
"We would never let them down unless it was totally unavoidable," he said.
The band had hoped to reschedule the dates, but it has not been possible because of the availability of the venues affected.
They said ticket holders for the cancelled gigs should apply for refunds at the point of purchase.
Duran Duran's All You Need Is Now tour has been under way since spring 2011.
They were forced to postpone a string of dates on the UK leg last summer after singer Simon Le Bon was treated for ongoing throat problems.

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19 June 2015 Last updated at 12:11 BST
Drone footage produced by Chester Unlocked has revealed hidden craftsmanship such as the Chester Imp carving high up in one of its windows.
Other iconic landmarks in the city will be filmed throughout the year.
Duran Duran have been forced to cancel the remaining two dates of their world tour because keyboard player Nick Rhodes is still too ill to perform.
[NEXT_CONCEPT]
A history project has given a glimpse of Chester cathedral which has not been seen for centuries.