Got into an argument over this. Someone insisted that a US Navy SEAL is a "soldier". This is just one of those things where I've encountered enough US Navy personnel to know that's they don't take kindly to be called soldiers. Marines want to be called Marines, and USAF are "airmen" (even women). Not sure if seaman or sailor is preferred, but I can't think of anyone who served in the US Navy who would object to being called a sailor.
All personnel in the uniform of naval service are Sailors first, and in addition they are officers, chiefs, petty officers, aviators, Seabees, surface warriors, and submariners. This is an important point impacting unity and esprit de corps.