Brody (and other US Marines) are referred to multiple times as "soldiers", including by a character who claims he was with a UDT (or underwater demolition team, a "sister" team to the Navy SEALs who have to pass the same training). In reality, it is a point of pride among US Marines that they are not soldiers, but marines. The Navy UDT character ought to know this as well. In turn, Brody refers to the UDT character as a soldier as well. As Naval personnel, he would be a sailor.
Brody's uniform ranking changes several times throughout season 1. At highest rank on service uniform he has the ranking of Gunnery Sergeant. In dress blues to attend dinner, he is shown with only a Sergeant ranking. In reference to repeated naming of ranking, he is referred to as Sergeant, if the Gunnery Sergeant rank is true, everyone would refer to him as Gunnery Sergeant or Gunny for short