The shooting took place in 2010 at the 180-foot border separation in a dry bed of the Rio Grande between the El Paso, Texas, and Ciudad Juárez, Mexico.
The family of Sergio Hernandez is seeking to sue the Border Patrol officer in the 15-year-old's death, arguing the agent violated Sergio's constitutional rights.
The shooting sparked outrage because Sergio was not armed and he was hiding behind a pillar about 60 feet away from the officer.
Court briefs filed by lawyers for the boy's family said Sergio and his friends were playing chicken -- daring each other to run up the incline on the U.S. side to touch the fence. A cellphone video of the encounter shows U.S. Border Patrol officer Jesus Mesa arriving on a bicycle and grabbing one of the kids at the fence on the U.S. side while the others ran away. Sergio ran past Mesa and hid behind a pillar on the Mexican side of the border beneath a bridge.
While holding down another child, Mesa fired three shots -- one of which hit Sergio in the head, killing him. Other U.S. border agents arrived shortly after.
As U.S. Border Patrol agents are prohibited from entering the Mexican side of the border, the officers soon left without providing medical assistance to Sergio, the family alleges.
The FBI's El Paso office said Mesa fired his gun after being "surrounded" by suspected illegal immigrants who "continued to throw rocks at him." Two days after the FBI's statement, the cellphone video surfaced, contradicting the FBI's comment that Mesa was surrounded.
In 2012, the U.S. Department of Justice declined to file charges in the shooting, saying civil rights violations did not apply because the teenager was not inside the United States when he was killed. The investigation also concluded the agent acted within U.S. Border Patrol regulations.
The question before the high court is whether the family of Mexican nationals has the right to sue for a violation of the U.S. Constitution.
Univision and CBS published the video of the shooting as part of a report in Spanish.