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Police on the island of Phuket told CNN the man killed the baby and then committed suicide by hanging himself. The two bodies were discovered hanging in a derelict building in the Sakhu area of the southern Thai island.
Police Maj. Prawat Tantibhussapun said the father's cousin was browsing Facebook on Monday and came across the live video in which the man, identified as 20-year-old Wutthisan Wongthale, was putting a rope around his daughter's neck.
Police arrived at the scene at 6:30 p.m., about an hour after the live feed started.
Police said the man had an abusive history toward his wife -- the day before he tried to change his wife's Facebook profile picture and they had fought.
She was threatened by him, Prawat said, so she left the family home and came back, too late, to find her daughter the next day.
Dozens of people, including the girl's mother, attended the baby's funeral Tuesday.
Two videos showing the child's death were posted on the man's Facebook page for about 24 hours, Group Capt. Somsak Khaosuwan from Thailand's Ministry of Digital Economy and Society told CNN.
He said the ministry coordinated with Facebook to remove the videos, and they were taken down Tuesday afternoon.
Facebook statement
"This is an appalling incident and our hearts go out to the family of the victim. There is absolutely no place for acts of this kind on Facebook," a company spokeswoman said.
That slaying prompted Facebook CEO Mark Zuckerberg to say at a developer's conference: "We have a lot of work, and we will keep doing all we can to prevent tragedies like this from happening."
CNN's Spencer Feingold and journalist Patrick Sarnsamak contributed to this report.