Karolina Dabrowska's son Sebastian was born with the umbilical cord wrapped around his neck. As his father Matthew Errington, struggled to get him breathing, the Addison Lee driver demanded the couple pay a fee for clearing up.
Errington, 43, said the driver then banged on the ambulance doors and demanded he either clean up himself or hand over cash.
"The actions of the taxi driver at the most stressful moment of my life was appalling," said Errington.
The couple live with Dabrowska's eight-year-old daughter in Peckham, south east London.
When Dabrowska, 36, went into labour they ordered a minicab from their home to University College Hospital, in Fitzrovia, central London.
But Dabrowska's waters burst on route and the driver pulled over outside Aldwych One hotel, near Covent Garden.
Luckily a doctor happened to be staying in the hotel, so they were able to help deliver baby Sebastian, while hotel staff gave Errington towels.
Errington said: "She said the baby's coming, we need to stop. So we pulled over. I dialled 999 for an ambulance and ran into the hotel.
"There was a doctor in there and two minutes later my son was born in the back of the taxi.
"My baby came out with his cord wrapped around his neck and at that point the taxi driver starts asking me for money.
"I said, 'My son's not breathing leave me alone'.
"The doctor came out and saved my son's life.
"Everybody was so helpful, except the taxi driver saying, 'How am I going to get my taxi clean?'
"I got my son breathing and he is still asking for money."
Sebastian Errington was born at 12.30am on October 25. His father paid the driver £40 for the fare, hoping that no "fouling charge", which can be up to £100, would not be charged.
But Errington said: "[The driver] started banging on the doors to get into the ambulance and said get out and clean the taxi."
Addison Lee chief executive Andy Bolan wrote to the family to apologise for his driver's behaviour, refunded the fare and made a £100 donation to Bliss, a premature baby charity.
A spokesman from Addison Lee said: "We would like to wholeheartedly apologise again for the service that Matthew and Karolina experienced, at what was a stressful time for them.
"As soon as we investigated what had happened we refunded their fares for both journeys to and from hospital, discussed with the driver how he could have been more sensitive to Matthew and Karolina's needs at a difficult time and are issuing guidance to all our drivers to ensure this unfortunate situation can never happen again.
"As a gesture of goodwill we have also made a donation to the premature and sick babies charity, Bliss."