no, some salt is mined
All salt is sodium chloride (NaCl), and it all comes from seawater — even table salt.
Table salt is typically mined from salt deposits, remnants of older bodies of seawater that have since dried up and are long gone.
Sea salt is crystallized from current bodies of seawater. Unrefined sea salt may contain impurities, which may or may not contribute to a more complex flavor. Of course, if you look at the amount of salt you actually sprinkle on a large grilled steak, for example, whether or not those trace impurities really contribute much flavor to your food is up for debate.
The real appeal with using "sea salt", as opposed to table salt, is that it sounds exotic.