They can try, but they have to go to court. In the mean time the point is made.
The Catholic Church is built on the slain bodies of Martyred Saints, both known and unknown to history. ALL the martyrs of the Church where murdered by GOVERNMENTS. Weather Nero Caesar, Robespierre, Hitler, Mao, or any other number of authoritarians.
The harder the despots squeezed the Church, the larger and more powerful the Church became, over time. As they have modified and acquiesced to the secular modernity, they have become less influential. They have lost members to other, more strict theologies, or to no theology at all. The more they are 'inclusive' the less relevant they become.
I understand and appreciate your point. I'm guessing though that the obama administration is more aware than most catholic laity that many of our hospitals have become secularized in recent decades. Just as our colleges have. Some claim it's due to accepting government money (Medicare, Medicaid, etc) but I don't buy that argument. It'd be illegal to discriminate against those patients anyway. No, there are many organizations that contract with the feds to provide a service and remain essentially true to their mission. I think what's happened in Catholic hospitals is that many of the religious orders that founded them got out of the health care ministry and turned them over to organizations that haven't remained committed to Catholic health care principles. If our institutions survive this mess, it'd be nice if some of those orders resumed their former health care ministries, at least at an administrative level to reinvigorate and oversee adherence to Catholic health care principles in these facilities.
Peace be with you.