Mr Cannon, 31, is spending the first week of the new year in hospital in the Colorado ski resort of Aspen, where he and his wife usually spend Christmas.
Miss Carey, 41, tweeted that Mr Cannon is suffering from "mild kidney failure."
She posted a picture on her website of a miserable-looking Mr Cannon in a hospital bed as she lay beside him.
She asked for prayers and said Cannon's situation was "very painful." She later called it a "serious moment that's very tough on all of us."
The pair became the parents of twins – a boy and a girl, Moroccan and Monroe – last year.