These paragraphs from the encyclical, Diuturnum Illud, On the Origin of Civil Power (1881) are particularly relevant to the situation facing Catholics in the U.S. today:

15. The one only reason which men have for not obeying [their rulers] is when anything is demanded of them which is openly repugnant to the natural or the divine law, for it is equally unlawful to command to do anything in which the law of nature or the will of God is violated. If, therefore, it should happen to anyone to be compelled to prefer one or the other, viz., to disregard either the commands of God or those of rulers, he must obey Jesus Christ, who commands us to “give to Caesar the things that are Caesar’s, and to God the things that are God’s,” (Mt 22:21), and must reply courageously after the example of the Apostles: “We ought to obey God rather than men” (Acts 5:29). And yet there is no reason why those who so behave themselves should be accused of refusing obedience; for, if the will of rulers is opposed to the will and the laws of God, they themselves exceed the bounds of their own power and pervert justice; nor can their authority then be valid, which, when there is no justice, is null.

16. But in order that justice may be retained in government it is of the highest importance that those who rule States should understand that political power was not created for the advantage of any private individual; and that the administration of the State must be carried on to the profit of those who have been committed to their care, not to the profit of those to whom it has been committed. (Emphasis added)Let princes take example from the Most High God, by whom authority is given to them; and, placing before themselves His model in governing the State, let them rule over the people with equity and faithfulness, and let them add to that severity which is necessary, a paternal charity. On this account they are warned in the oracles of the sacred Scriptures, that they will have themselves some day to render an account to the King of kings and Lord of lords; if they shall fail in their duty, that it will not be possible for them in any way to escape the severity of God: “The Most High will examine your work and search out your thoughts: because being ministers of his kingdom you have not judged rightly...Horribly and speedily will he appear to you, for a most severe judgment shall be for them that bear rule...For God will not accept any man’s person, neither will he stand in awe of any man’s greatness; for he made the little and the great, and he hath equally care of all. But a greater punishment is ready for the more mighty” (Wis 6:4-6, 8-9).

26. The Church of Christ, indeed, cannot be an object of suspicion to rulers, nor of hatred to the people; for it urges rulers to follow justice, and in nothing to decline from their duty; while at the same time it strengthens and in many ways supports their authority. All things that are of a civil nature the Church acknowledges and declares to be under the power and authority of the ruler; and in things whereof for different reasons the decision belongs both to the sacred and to the civil power, the Church wishes that there should be harmony between the two so that injurious contests may be avoided. ...