The DJ was due to host Tuesday's Radio 5 live Breakfast show with Rachel Burden but became unwell at home on Monday night with the painful condition.
Campbell, also a presenter on Long Lost Family, told fans the attack was the "worst pain ever" and praised the "amazing care" he received.
He tweeted his fans: "Good morning from Salford Royal Hospital where I was rushed last night after a kidney stone attack. Blimey. Some pain that!
"Amazing care from compassionate professionals."
Responding to one well wisher he said: "Worst pain ever. I tell ya. Pain relief was excellent."
While another follower quipped that the illness was "some length to go to get out of Brexit chat", Campbell responded: "Haha. Stop it. Hurts to laff [sic]."
DJ Chris Warburton was called in to replace Campbell and presented Tuesday's show alongside Burden.
Burden told listeners she had been in touch with her regular co-host and he was recovering well.
Campbell has presented the show since 2003 and last year confirmed on air that he was taking a pay cut amid the controversy over the gender gap.
BBC Radio 5 live Breakfast airs every morning from 6-9am.
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