The fighting happened on once Mexican land that now constitutes the cities of Pico Rivera, Montebello and Whittier.
Rebollar Gomez's son, 2nd Lt. Gibram Cruz is not allowed to travel to foreign countries without permission from the military.
Mexico's President Andres Manuel Lopez Obrador said there is "great cooperation" to fight drug trafficking.
The award is designated for troops who "encounter no foreign armed opposition or imminent hostile action."
Marine and Naval Criminal Investigative Service officials have not released any details on the pending charges.
One thousand Texas National Guardsmen will joint 1,100 active-duty troops on the border in the next several weeks.
Two Marine infantrymen were pulled over and arrested by U.S. Border Patrol, along with three undocumented immigrants.
The government has faced criticism for stopping migrants from crossing the US-Mexican border.
U.S. service members have helped apprehend more than 100,000 illegal immigrants, the U.S. Border Patrol's chief says.
Border Patrol spokesman Carlos Pitones said troops on Friday will begin painting a one-mile section in Calexico, California.