An oath began as a mystical thing, and I think retains a bit of that quality to it even today. It is possible for an atheist to have a commitment that is of an almost religious quality.
What does that even mean in this context? Why should that mean this man should be forced to invoke a god he doesn't believe in?
I would think it means he shouldn't have to, as it would contradict his religious commitment. I wonder if there could be some other source of help he can reference that is more in line with his beliefs.