Homicide squad Detective Senior Constable Kevin Michael Burke said emergency services were called to a Glenroy address at about 8.30am on Sunday.
It is alleged the accused was caring for the child, while the mother was shopping.
When the boy became unconscious, the accused contacted the mother of the child, who asked a friend to ring the ambulance.
Emergency service workers found the accused walking out of the house before showing them to the scene, where they found the baby boy unresponsive.
The baby, Chayse Dearing, was taken to the Royal Children’s Hospital in a critical condition with head injuries, genital injuries and ligature marks around the neck.
Senior Detective Burke told the court medical professionals had deemed the child’s injuries to be “unsurvivable” and decided to “withdraw” life support.
He told the court the homicide squad located Mr Lindsey after appeals to the public, and arrested him on Thursday in Sunshine.
Senior Detective Burke said Mr Lindsey was questioned, but made “no comments” during the interview.
Bail justice Claire Vissenga asked Mr Lindsey if he was on any medications.
Mr Lindsey said he was, but had not told the police about it.
Ms Vissenga asked the police to make a note and facilitate the medications.
Mr Lindsey was refused bail and remanded in custody. He will appear in the Melbourne Magistrates Court on Friday morning.