Officers found the bodies of 20-year-old Wuttisan Wongtalay and the baby in an abandoned hotel in Phuket province after receiving reports of the video.
Police Col Jirasak Siemsak said Wuttisan and his wife had been arguing.
The man made the broadcast on Monday evening, and the video was made inaccessible by Facebook late Tuesday afternoon.
The incident came less than two weeks after a man in Cleveland, Ohio, in the United States broadcast himself on Facebook Live fatally shooting a man.
A series of other offensive broadcasts, including sexual assaults, has caused Facebook to seek ways to block such videos as quickly as possible.