I do not know to which point you have separation of church and state, taken that the oath is done on the Bible (at least traditionally), with a formal "So help me God" at the end (https://en.wikipedia.org/wiki/Oath_of_office_of_the_presiden...). It sure can be omitted and you could swear on something else than the Bible but this sends a message.

We French do not have a clear cut separation of the two either, but this is more a cultural thing because a minimal percentage of the population is religious, at least with the religion our culture was raised with (Christianity, roughly speaking).

Also, when we passed in 1975 the abortion law (Loi Veil), the opposition was not for religious reasons but, sadly, a pure masculine view on the role of the woman and her rights.