Can my french public university require me to have a Google account

I would keep this practical and not make the professor the first legal battlefield. Ask in writing for an equivalent alternative that does not require a personal Google account, and copy/ask the course coordinator, student support, or the university DPO if you have one. In France a public university should at least be able to tell you the legal basis and what alternative exists for students who cannot or do not want to create a third-party account. If you already made the account, you can still ask for next time: keep it empty, use it only for that class, and delete/export it after the course if the university confirms you no longer need it.