Wigan, currently 23rd in the Championship, have won just two league games this term and the decision was made to end Caldwell's 18-month stint in charge at the DW Stadium.
Sharpe, who took over as chairman from his grandfather Dave Whelan in March last year, told the club's official website: "It's undoubtedly the toughest decision I've had to make since becoming chairman.
"We deservedly won the League One title last season. However, we have not managed to carry on that momentum into this season and, after careful consideration over a period of time, I feel that we need to act now in the best long term interests of the club."
Former Scotland defender Caldwell, who ended his playing career at Wigan and captained the club to FA Cup success in 2013, was appointed boss in April 2015.
However a run of one win in 12 Championship games leaves the Latics facing an immediate return to League One.
"We thank Gary for all the hard work he has put in and wish him all the best for the future," added Sharpe.
"We need to identify a new manager who can have an immediate impact and deliver success in the long term."