Frances Cappuccini, a 30-year-old primary school teacher, died at Tunbridge Wells Hospital in 2012.
Dr Errol Cornish, 67, of Bromley, south-east London, faced a charge of manslaughter by gross negligence at the Old Bailey.
Maidstone and Tunbridge Wells NHS Trust denied corporate manslaughter.
Mr Cornish, of Holmbury Park, and Maidstone and Tunbridge Wells NHS Trust, which runs Tunbridge Wells Hospital, appeared before Mr Justice Coulson.
It is the first time an NHS trust has been charged with corporate manslaughter since the offence was introduced in 2008.
A trial has been set to start on 11 January 2016 at the Inner London Crown Court.
A doctor has pleaded not guilty to the manslaughter of a woman who died giving birth to her second son by emergency Caesarean.