The 29-year-old said the infection was something “I wouldn’t wish on anyone” and had been instructed by doctors to “rest and recover”.
Payne was part of the original line-up of the world-famous boy band, formed in 2010, alongside Zayn Malik, Louis Tomlinson, Niall Horan and Harry Styles.
In an Instagram post on Friday, Payne wrote: “It’s with a heavy heart I have to tell you that we have no other choice but to postpone my upcoming tour of South America.
“Over the past week I’ve been in hospital with a serious kidney infection, it’s something I wouldn’t wish on anyone, and doctors’ orders are that I now need to rest and recover.
“I was beyond excited to come play for you guys.
“To all of you who have bought tickets: I’m so sorry. We’re working to re-schedule the tour as soon as we possibly can, but for now we will be refunding the tickets – so please look out for updates from your point of purchase.
“Thanks as always for the love and support, and look forward to seeing you soon.”
In an accompanying video, Payne added that he had just begun rehearsing for the tour before being told that “now was really not the right time to be out on the road”.
He promised fans that once recovered, he would return with “an even bigger, better show.”