The question is: would the school district have granted unpaid time off to the Catholic teacher? If not, then they aren't discriminating.
Employers (public and private) are required to make "reasonable accommodation". If someone takes a job that plainly requires they are expected to work the 180 days of the school year, then they must do so unless a personal or family emergency makes it impossible.
Some school districts give teachers a handful of "personal" days each school year that can be used at the teacher's discretion. But at least in my local school district, the teacher is required to find a substitute teacher for planned absences.
Other than those personal days, teachers are limited to the same holiday/vacation periods as students: summer, spring break, Thanksgiving, Christmas, etc. They can't take lengthy vacations in the middle of a school year outside those planned holidays.
“They can’t take lengthy vacations in the middle of the school year...”
You are absolutely correct. A substitute teacher for a MONTH would disrupt any sort of education process. This teacher has the summer months to do what she likes. If a visit to Mecca is required Once In a Lifetime... you can wait till she retires or do it over the summer. Just a thought.