No—registering an entity with a state’s Secretary of State does not prevent the name of that entity from being used by a different entity in a different state, nor does that act alone offer trademark or service mark protection for the name.
When an entity wants to establish a national reputation, it should conduct a broad name search through trademark, trade name and service mark registries before making a significant investment in the name. Also, the entity should check the availability of its corporate name (or variations of the name) with Internet domain name registries.