Militant groups aren't wielding 'laïcité' in support of a public sphere embracing diverse religious expression. The employ it as a form of anti-Islamic politics.
Making sure late-term abortions are done only to save the life of the mother—under the safest conditions possible—should be something both sides can agree on.
If you doubt the cost of indulging in political pieties rather than political organizing, compare the influence of Occupy Wall Street with the Tea Party.