Stewart's wife Lynn has been fighting cancer since 2013 and, as she begins a new course of treatment, he has decided to temporarily step away from his day-to-day duties at Surrey to care for his wife and family.
In a statement issued on his behalf by the club, Stewart asked for privacy as he takes a break from cricket. Steve Elworthy, the club chief executive, will assume his duties on an interim basis.
Elworthy said: "All of our thoughts are with Alec, Lynn and the family and we wish them the very best. The club will fully support Alec and the family through this leave of absence and ask everyone to please respect his request for privacy."
Stewart, 59, has been in his role at Surrey since 2014, and last season helped guide the club to the County Championship, having also won the title in 2018.
In the aftermath of last winter's Ashes loss, Stewart had been among the favourites to take over from Ashley Giles as England's director of cricket - a role that subsequently went to Rob Key - but he pulled out of the running for both that job and the national selector role due to his family commitments.