When she was rescued, baby Aya - meaning miracle in Arabic - was still connected to her mother by her umbilical cord.
Her mother, father and all four of her siblings died after the quake hit the town of Jindayris.
Aya is now in hospital.
"She arrived on Monday in such a bad state, she had bumps, bruises, she was cold and barely breathing," said Hani Marouf, the paediatrician looking after her.
She is now in stable condition.
Videos of Aya's rescue went viral on social media. Footage showed a man sprinting from the collapsed debris of a building, holding a baby covered in dust.
Khalil al-Suwadi, a distant relative, who was there when she was pulled to safety, brought the newborn to Dr Marouf in the Syrian city of Afrin.
Thousands of people on social media have now asked for details to adopt her.
"I would like to adopt her and give her a decent life," said one person.
A Kuwaiti TV anchor said, "I'm ready to take care of and adopt this child... if legal procedures allow me to."
The hospital manager, Khalid Attiah, says he has received dozens of calls from people all over the world wanting to adopt baby Aya.
Dr Attiah, who has a daughter just four months older than her, said, "I won't allow anyone to adopt her now. Until her distant family return, I'm treating her like one of my own."
For now, his wife is breastfeeding her alongside their own daughter.
In Aya's home town of Jindayris, people have been searching through collapsed buildings for loved ones.
A journalist there, Mohammed al-Adnan told the BBC, "The situation is a disaster. There are so many people under the rubble. There are still people we haven't got out yet."
He estimated that 90% of the town had been destroyed and most of the help so far had come from local people.