If the girls instead swore an oath to obey god, what would that mean? Which god? Christian? OT or NT? Hindu? Xenu? Thor? And to obey which parts of what that god said?
Besides, saying something does not mean a person believes it. Like every psychopath who swore to tell the truth on a bible really told the truth? "I swear I did not kill, chop up and eat that family. But they were tasty."
Why do religious people think that having religion around makes people good?
The oath might as well be bland, so at least it includes everyone.