Police dashcam video shows Danny Ray Thomas, 34, walking towards the officer on a road in Houston, Texas, last Thursday before a gunshot rings out.
Deputy Cameron Brewer repeatedly shouts “get down on the ground” before firing. Both the officer and victim are black.
Mr Thomas’ family say he suffered from depression after his children died.
His sister, Kita Thomas-Smith, attended a Houston City Council meeting on Tuesday where she tearfully urged politicians do to something about “police brutality”.
Mayor Sylvester Turner told her: “I certainly feel your pain. We are certainly sorry for any life that is taken.”
Outside city hall, Ms Thomas-Smith told the Houston Chronicle of the dashcam video. “It feels like premeditated murder.
“He was clearly walking, not running, toward the deputy not trying to hurt or harm him.”
Deputy Brewer, who had a Taser with him, has been placed on administrative leave as police investigate the shooting.
Houston Police Department said in a statement last week that Mr Thomas was found “walking in the middle of the intersection” of a busy road in the city.
Deputy Brewer noticed him with “his pants around his ankles, talking to himself and hitting vehicles as they passed by”.
“Thomas then struck a white vehicle, and the driver exited and engaged in a physical altercation with the suspect,” Houston police said.
The Harris County Sheriff’s deputy saw the fracas and stopped his car to intervene.
“Fearing for his safety, the deputy discharged his duty weapon, striking Thomas once in the chest,” said Houston police, which is leading the investigation.
In a different video taken by a bystander and published by the Chronicle last week, onlookers at a bus stop are heard laughing as they predict the officer will use a stun gun on the jaywalker.
“He’s about to get tased,” says Kaaryn Young as she films.
Gunfire rings out.
She is heard asking in disbelief: “He shot that man?
“He should have gotten tased!
“He shouldn’t have shot that man in the street.”
According to relatives, Mr Thomas’ partner murdered his two children in 2016.
She is awaiting trial for the deaths, which happened while Mr Thomas was in prison serving a three-year drug-related sentence. (Courtesy BBC)