The 55-year-old Yorta Yorta woman was placed in a Castlemaine cell after being arrested for drunkenness on a train on December 5, 2017.
She died two weeks later.
"I am deeply sorry for your loss and deeply sorry for our disrespectful care and treatment Tanya received that day," Ambulance Victoria clinical operations director Michael Stephenson said to Ms Day's family during an inquest on Tuesday.
"It was a dark moment for our organisation," he later said.
"I'm sorry."
Members of Ms Day's family thanked Mr Stephenson for the apology.
Police officers did not see Ms Day fall or enter the cell until 8.03pm, when they noticed a bruise on her forehead and called an ambulance.
The first paramedic on the scene was Lisa Harrup, who said she was told by a police officer he saw Ms Day slip from a seated position.
"All of my treatment was based on what I thought was an accurate history provided by a police officer. Why wouldn't he know exactly what had happened?" she told the inquest last week.
"It was upsetting to me that I might have missed such an important part of a diagnosis."
Tuesday's evidence in Melbourne coincided with a rally that marched from state parliament to the Coroners Court to show support for Ms Day and her family.
The inquest didn't sit on Monday as Sunday would have been Ms Day's 57th birthday.