The 52-year-old British musician, keyboardist and composer was touring as part of The Smile with fellow Radiohead singer Thom Yorke as well as Sons of Kemet drummer Tom Skinner before being forced to cancel the supergroup's European leg on Friday.
A statement from The Smile said Oxford-born Greenwood had been "seriously ill" and would "soon return home", but added doctors had advised he needed rest.
A statement posted by The Smile on X, said: "A few days ago, Jonny became seriously ill from an infection that needed emergency hospital treatment, some of it in intensive care.
"Mercifully he is now out of danger and will soon return home.
"We have been instructed by the medical team in charge of Jonny's care to cancel all engagements until he has had time to make a full recovery.
"To that end, the smile tour of Europe in August is cancelled.
"We all wish Jonny a speedy recovery."
The band has confirmed all ticket holders will be refunded at the point of purchase.
The band has released two UK top 10 albums in A Light For Attracting Attention and Wall Of Eyes.
The Smile's sound has been compared to Radiohead, but with more of an influence from Krautrock, a form of German experimental music, and jazz.
The band was set to play venues in Switzerland, Sweden, Germany and Poland next month.
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