Saying “Bless you!” after a sneeze is not even very religious. In any case, it is not sectarian.
Should we also be banned from saying “Good-bye”? That is probably more religious, because it literally means “God be with you” (God by ye).
This calls to mind the incident recounted by Roy Campbell in his autobiography “Light on a Dark Horse.” He tells of a young recruit in the leftist “Republican” forces in the Spanish Civil War, who was brought before his commander for violating the order that all religious greetings be replaced by “Salud” (health).
After a lecture admonishing the young soldier, and warning of punishment for future infractions, he waved him away, saying, “so let’s hear no more of this! Adios!”
Adios, of course, is short for “go with God.”