Andrew Tahmooressi, a 25-year-old veteran of the war in Afghanistan, was arrested on March 31 when he entered Mexico from California.
His family has said that Tahmooressi suffers from PTSD and crossed the border inadvertently with several weapons in his car that he had purchased legally in the United States.
"I am respectfully requesting that you personally engage the Mexican government and use the resources of the Department of State to intervene on Andrew's behalf to resolve this matter quickly so that he can return home," Scott said in a letter to Kerry that was made public.
"Andrew served our nation in two tours of Afghanistan, and it's incumbent on us all to resolve the situation in the timeliest manner possible," the Republican governor said, criticizing the State Department's "inaction," which he called "frankly unsettling."
Several US lawmakers have requested the release of the former marine.
Republican Representative Ed Royce, who chairs the House Foreign Affairs Committee, and other congressmen visited Tahmooressi last week in the El Hongo prison in Tecade, and posted a photo of the encounter on Twitter.