Aysenur Ezgi Eygi, 26, was shot in the head as she took part in a demonstration in the Palestinian village of Beita in September.
Israel has said it is investigating her death, describing it as "unintentional" from a ricocheted bullet, but the family says this claim is contradicted by the available evidence and eyewitness accounts.
After meeting Mr Blinken in Washington on Monday, Ms Eygi's widower Hamid Ali said the family was "not optimistic" that justice would be served.
"He was very deferential to the Israelis," Mr Ali told the BBC. "It felt like he was saying his hands were tied and they weren't able to really do much."
State Department spokesman Matthew Miller said Mr Blinken offered the family his "deepest condolences" during the meeting.
"It was a death that never should have happened," Mr Miller said. "He told them that Israel has told us in recent days that they are finalising their investigation."
Pressed by the BBC over the family's dissatisfaction with the US response, Mr Miller said: "We will demand answers from the government of Israel."
BBC