From a libertarian perspective, public education system is not desirable in a free society, roughly because public education is a monopoly (in this case, of the State), which is based on the compulsory extraction of everyone's income and to use it to make children to be educated according to a curriculum set by the government. For this reason, private education and home schooling are regarded as the main systems to train and educate children, and the parent's freedom of choice when it comes to their children's education should be respected.
Since in a free society (again, from a libertarian perspective) can exist many different types of privately owned education systems, and parents can freely choose the schools that they regard as the best for their children education without government interference, they could make use of their liberty and choose to send their children to private religious schools.
What if these schools teach certain kind of ideologies, such us the denigration or disapproval of homosexuals or certain ethnic minorities based on religious principles? Would it be justifiable for a libertarian state to limit parent's free choice by means of imposing control or even banning some kind of religious schools?
I think that if these institutions taught explicit hate ideologies against certain people, it would be easier to answer, but since the limit between teaching "hate" or just "soft contempt based on religious and moral principles" could be very permeable, the question can be difficult to answer.