The Gingerbreads sit 12th in the EVO-STIK League Premier Division, and have lost five consecutive games.
Their 3-2 reverse at home to Nantwich Town on Tuesday night proved to be the last straw for Culverhouse, who the club confirmed has resigned with immediate effect due to personal reasons.
Chairman Nigel Wedgwood said: "It is with regret Grantham Town Football Club accepts the resignation of Ian Culverhouse. “On behalf of everyone at the club, may I express our thanks for his hard work and commitment in his short time here.
“Ian leaves the club on best of terms and we very much wish him well in the future."
Grantham reached the play-off final last season and were beaten by Ashton United, who earned promotion to National League North.
They started October with a home loss to Hednesford and a 5-0 defeat to South Shields, and have also been beaten by Buxton and Lancaster this month.
They suffered an early FA Cup exit when they lost 2-0 at home to St Ives Town in a second qualifying round replay, and were also beaten 6-0 at home by Glossop in the Integro League Cup.
Keep up-to-date with our exclusive email newsletters.
Subscribe