Theoretically, yes. Realistically, if anyone tried, there would be a long line of people wanting to talk with that person that tried to deny sick leave, and the first ones in line would be HR.
Vacation is more likely to be denied, but I've never had it happen and have never denied any myself.
This is a bit more insidious than it appears- because it's asking kids to rat out atheists to religious officials.
There are absolutely areas in the US where being discovered as an atheist will ruin your life.