Many people believe everyone supporting a policy should have a secular reason to do so. Deferring, for the moment, the question if that is a sensible expectation, I wonder how we decide whether an argument is secular.
I acknowledge one fairly obvious category of non-secular arguments: Something knowable only through a special religious revelation is clearly tied up in the religion that claims that revelation. So, for example, "it's in the bible" is not a good reason to impose something on people who don't believe the bible to be God's word. Now that would be a significant concession if I was a divine command theorist, but the thing is I'm not. I believe most commandments we know through scripture and tradition can also be derived from the natural law. And those not so derivable mainly relate to religious ritual non-Christians don't participate in anyway. This is not something I made up yesterday, Catholics have been thinking this long before anyone thought of not having an established religion.
So suppose I say the state should never pay for anyone's artificial contraception because artificial contraception separates the unitive and the procreative purposes of sex and is thus immoral. Does that count as a secular argument? If so I have no objection to being limited to secular arguments. But I think most people concerned about religious arguments in politics would say it doesn't. So being based on special revelation is not what people objecting to religious arguments in politics mean by an argument being religious. Rather, they would claim that both objective purposes of human actions and a principle against undermining them are fundamentally religious ideas. But why would they be religious when the equally non-empirical idea of the equality of all humans counts as wholly secular?
Well, as long as nobody can give me a less cynical definition I think an argument gets called religious if and only if it fulfills the dual conditions of (a) being a part of our civilization's traditional morality, which of course was Christian and (b) now being controversial.
But then excluding "religious" arguments is a recipe for getting rid of any moral principle we now agree on that should later get controversial.
For example, reactionaries like me believe human life is sacred even if the human doesn't want to live anymore. Thus no assisted suicide. The modern agnostic liberal quite simply doesn't believe that and thus decides by the principle of autonomy which is still uncontroversial/secular. Now we reactionaries think this will put us on a slippery slope to killing people against their wills. This again seems far-fetched to agnostic liberals; after all consent is a very obvious bright line. But suppose at some future date a significant minority believes consent isn't that important and we can alleviate suffering by non-consensual euthanasia. And then they ask us why consent should be morally relevant. Because people have autonomy or dignity or something? - Says who? Because there is a right to life? - Why should there be? Next you'll tell us it's because people are created in the true image and likeness of God! So just get out of our way and let us decide by secular principles like pain reduction. Seriously now, you're totally allowed to hang on to some residual religious ideas, just keep them out of politics.
Now you might claim the right to life is a secular idea and that is true now. But how can it still be true in a society it is not universally accepted in? If you find a criterion that makes it secular, how do you escape making Catholic moral law theory secular too?
So basically either natural law is a secular argument, or secular arguments can only ever disestablish moral principles and never establish new ones. In the latter case the only end continued secularism can progress to is the Hobbesian state of nature. That surely isn't a reasonable plan for moral progress.