Whatever one may say about the mandate, the accommodation, or Catholic views of the matter, this argument is a rather bad argument for the simple reason that the two claims are not logically incompatible with each other. The two claims mentioned, again are:
Most critics of the HHS contraception mandate have said the controversy is about religious liberty, not contraception. Some of the same critics have said that the question is not whether Catholics could comply with the mandate in good conscience if they had to, but whether the government ought to force them to comply with it in the first place. But these two claims are logically incompatible with each other....
But if the argument is about religious liberty, then critics must persuade those who aren’t opposed to contraception or the coverage mandate that requiring Catholic employers to provide such coverage—or facilitate it in any way—would force them to violate the teachings of their religious community. If it can be shown, therefore, that such requirements would not force Catholics to violate their church’s teachings, then no one can oppose the mandate on the grounds of religious liberty.
(1) The controversy is about religious liberty, not about contraception.
(2) The controversy is about whether the government ought to force Catholics to comply with the mandate, not about whether they could comply with the mandate in good conscience if they had to.
The contrastive clauses in each case simply deny an intentional object of the controversy; whereas the main clauses identify an intentional object. For these to be logically incompatible, the following claim would have to be self-contradictory:
(3) The controversy is about religious liberty and whether the government ought to force Catholics to comply with the mandate.
And obviously this is not self-contradictory.
Of course, what Boudway is trying to do is combine the first positive clause with the other negative clause:
(4) The controversy is about religious liberty, not about whether they could comply with the mandate in good conscience if they had to.
And he wants to say that this is logically contradictory. It is clearly not, though. If the government passed a law forbidding attendance at Mass on weekdays, Catholics could comply with this in good conscience, but it would be nonsense to say that this has nothing to do with religious liberty. For that matter, since the precepts of the Church only bind to the extent that it is genuinely feasible to obey them, if the government forbade attendance at Mass entirely, Catholics could comply with this in good conscience if they had to, although they would also have a duty not to comply if they could manage that. And, indeed, many other religions are far more generous about what falls within this classification than Catholicism. With full sanction of Quran and hadith, for instance, Muslims could in good conscience comply with an extraordinary amount if they had to -- but that doesn't mean that they would not fight any external attempt to restrict their religious practices only to those things unequivocally commanded and forbidden by the Quran. And if they did fight it, as far as religious liberty rights go they would be entirely right to do so.
The claims Boudway considers could only be logically incompatible if nothing were a matter of religious liberty except what your religion provably demanded that you do (or not do). But this is a leap at best. Thus Boudway's conclusion,
Simply is not supported by his argument -- and is, in fact, obviously wrong. Showing that it is illict according to the Church's own teaching is merely the simplest and easiest way to show that enforcing it endangers religious liberty. Religious liberty questions are simply not as cut and dry as Boudway wants to suggest; they are complicated and difficult to negotiate, which is why liberal societies have often espoused the principle, if not always the practice, of not predetermining what can and cannot be a matter of religious liberty.
But if one is going to claim that the mandate violates the religious liberty of Catholics, one really does have to demonstrate that the material cooperation with contraception it requires is illicit according to the church’s own teaching.