Fugard, 46, called for Hampshire Hospitals to suspend Labour Line until it could be established what went wrong and questioned whether there was a money-saving motive behind the service.
He told the Telegraph that from early in the pregnancy, the couple's third, the trust had appeared "very keen" for he and his wife, 38, to consider having a home birth.
"It was seriously traumatic - I can't stress what a hell it was," he said.
"We were cut loose.
"They could hear down the phone what was happening but they didn't have sufficient humanity to send someone.
"If anything had gone wrong we wouldn't have had any idea what to do.
"We were just left sitting on the bed holding this grey, seemingly lifeless baby for an hour before an ambulance turned up."
Fugard, who owns an organic drinks business, said at one stage during his wife's labour, which lasted less than an hour, the Labour Line operator put him on hold.
However, the operator later admitted carrying on listening as Bryony Fugard, who works in marketing, gave birth.
He said: "When I picked up the phone again to tell her I'd just delivered a baby, she said 'I know, we heard'."
Hampshire Hospitals has since confirmed that, despite the Fugard's home lying just within Berkshire, the address is included in the trust's catchment area.
During the pregnancy, the couple had several consultations with the Basingstoke and North Hampshire Hospital midwifery team, whom they described as "lovely".
A midwife also visited them at them at home 10 days before the birth.
Fugard said that, regardless of the call operator's administrative error, "basic morality" should have prompted her to send help.
"It would only have taken one complication for there to have been a serious problem," he said.
Despite the traumatic birth, the baby girl, Dorothy, is healthy and "very pretty", her parents report.
Mary Edwards, Chief Executive of Hampshire Hospitals Foundation Trust, said: "We have looked after Mrs Fugard through her antenatal period and are very sorry that we did not provide the support we should have done when she was in labour.
"We are in communication with the family and have apologised for the errors we made.
"We are investigating what went wrong and we will share the findings with Mr and Mrs Fugard at the earliest opportunity.
"We will also make sure we learn the lessons from this to avoid it happening in the future."