Politically, discrimination against a minority is easy and generally needs nearly no justification. Usually you need strong arguments why protecting the particular minority is worthwhile, not the other way around.
So the question is, are there convincing reasons to make it politically infeasible to ban Muslim women from wearing body and face coverings?
There is going to be some xenophobics who will support oppression of Muslims. You are going to get some support from Feminists who hold that such body coverings are sexist patriarchal oppression of women.
Popular portarial (accurate or not!) of women wearing full-body coverings as being forced to do so under threat of violence from male family members makes the Feminist argument stronger, and would sway people who don't consider themselves particularly Feminist.
France has a "secular" tradition; their republic is rooted not only in freedom of religion, but also freedom from religion in the public sphere. It is illegal in france to wear a conspicuous cross in their public schools.
Legally, we have the court decision right here that uphend the constitutionality (at the EU level) of the french "Burqua Ban".
The Court examined the applicant’s complaints under Articles 8 and 9, with emphasis on the latter.
While personal choices as to one’s appearance related to the expression of an individual’s
personality, and thus fell within the notion of private life, the applicant had complained that she was
prevented from wearing in public places clothing that she was required to wear by her religion, thus
mainly raising an issue with regard to the freedom to manifest one’s religion or beliefs.
The Court found that there had been a “continuing interference” with the exercise of the applicant’s
rights under Articles 8 and 9, as she was confronted with a dilemma: either she complied with the
ban and thus refrained from dressing in accordance with her approach to religion, or she refused to
comply and would face criminal sanctions. The Court further noted that the limitation in question
was prescribed by the Law of 11 October 2010.
So the court found that her rights where restricted by the law.
The Court accepted that the interference pursued two of the legitimate aims listed in Articles 8 and
9: “public safety” and the “protection of the rights and freedoms of others”.
As regards the aim of “public safety”, the Court noted that the legislature had sought, by passing the
Law in question, to satisfy the need to identify individuals in order to prevent danger for the safety
of persons and property and to combat identity fraud. It considered, however, that the ban was not
“necessary in a democratic society” in order to fulfil that aim. In the Court’s opinion, in view of its
impact on the rights of women who wished to wear the full-face veil for religious reasons, a blanket
ban on the wearing in public places of clothing designed to conceal one’s face could be regarded as
proportionate only in a context where there was a general threat to public safety. The Government
had not shown that the ban introduced by the Law of 11 October 2010 fell into such a context. As to
the women concerned, they were thus obliged to give up completely an element of their identity
that they considered important, together with their chosen manner of manifesting their religion or
beliefs, whereas the objective alluded to by the Government could be attained by a mere obligation
to show their face and to identify themselves where a risk for the safety of persons and property
was established, or where particular circumstances prompted a suspicion of identity fraud.
So the claim that "it is about face coverings" was considered by the court and rejected. The government attempted to justify the ban as being "no face coverings", but the court conisdered that not to be the reason for the law, and also the law was not an efficient way to advance the interest the government was claiming to advance. Due to the fact that the rights of the woman where interfered with, this was not considered sufficient justification.
As to the “protection of the rights and freedoms of others”, the Government referred to the need to
ensure “respect for the minimum set of values of an open democratic society”, listing three values in
that connection: respect for gender equality, respect for human dignity and respect for the
minimum requirements of life in society (or of “living together”). While dismissing the arguments
relating to the first two of those values, the Court accepted that the barrier raised against others by
a veil concealing the face in public could undermine the notion of “living together”. In that
connection, it indicated that it took into account the State’s submission that the face played a
significant role in social interaction. The Court was also able to understand the view that individuals
might not wish to see, in places open to all, practices or attitudes which would fundamentally call
into question the possibility of open interpersonal relationships, which, by virtue of an established
consensus, formed an indispensable element of community life within the society in question. The
Court was therefore able to accept that the barrier raised against others by a veil concealing the face
was perceived by the respondent State as breaching the right of others to live in a space of
socialisation which made living together easier. It added, however, that in view of the flexibility of
the notion of “living together” and the resulting risk of abuse, it had to engage in a careful
examination of the necessity of the measure at issue.
This is where the court found the governments case convicing. The constitution permits the government to pass laws enable life in society "living together", and found that the governments interest in this case was sufficient to restrict the rights for people to wear religious face coverings.
It then judged the balance, and basically said that France has a wide margin here, and let them keep the ban. They cited the lack of otherwise universal agreement among member states as one of the sources of that wide margin.
2 judges (Nuβberger and Jäderblom) disagreed with this.