Correct, non sick leave is usually considered discretionary time off; meaning, it’s at the discretion of the manager to approve it.
That said, this manager sounds like a nut job. It’s legal to be a hard ass, but people don’t have to remain working for you.
Worst comes to worst, they could ask for a doctor’s note. And doctors notes are always pretty generic and basically say “yup, they can’t work.” But if the manager does ask for Dr’s notes, they need to apply the policy equally to all employees.
It’s usually a HIPAA violation if an employer asks for specifics about a medical issue.
Source: I’m a people manager who has had to go through a bunch of trainings about these laws.