The boy, named as Julius Czapla, was found seriously injured after emergency services were called to a block of flats in Muirhouse Place West around 9:30am on Saturday. He died a short time later.
A 40-year-old man today appeared at Edinburgh Sheriff Court following the death of the child.
Lukasz Czapla appeared from custody during a private hearing where he faced a charge of assault.
He is also facing road traffic charges including allegations of dangerous driving, driving while over the drink drive limit and failing to stop.
Czapla, from Edinburgh, made no plea and the case was continued for further examination and he was remanded in custody.
In the past few days, floral tributes, photographs, cards and teddy bears have been left by a tree in Muirhouse Place West by heartbroken loved ones, friends and neighbours.
One message reads: “Sleep tight little man, may you now rest in peace.”
Another read: Taken far too soon. RIP little one. Fly high with the angels xxx.”
Another, written in Polish, translated as: “We love you… forever in our hearts.”
Hundreds of tributes have also been left online over the past few days.
Local councillors have also expressed their condolences to those affected by the tragedy.
Almond ward councillor Graham Hutchison said: “This is a heartbreaking incident which has caused great distress in the community. My thoughts go out to the family and we are grateful to police for reassuring residents in that area at this difficult time.”
Another local councillor, Kevin Lang, said: “What it’s again shown is what a supportive community Muirhouse is. This is a place that has, unfortunately, had tragedies not dissimilar to this before and what we have seen is an outpouring of sympathy and support for this family.”