The Church's relationship to the State has never been an absolute "No," and has never been and absolute "Yes." The balanced response which both Scripture and Christian history provide is based on the Church's acknowledgment that legitimate authority comes from God, and that at the same time, the Church is a divinely-established entity which has a mission and an existence which transcends that of the State.
In his first letter, St. Peter expresses the "yes" of the Church to the State in the following passage: "Because of the Lord, be obedient to every human institution, whether to the emperor as sovereign or to the governors he commissions…Such obedience is the will of God" (cf. 1Pt.2:13-17).
Our Lord Himself said that we should "give to Caesar what belongs to Caesar, and to God what belongs to God" (Mt. 22:21) In that statement, we also see the "no" of the Church to the State. We must give God what is His. The coin belongs to Caesar because it bears the image of Caesar. What, then, belongs to God? That which bears the image of God -- which is human life, including Caesar himself. Caesar, then, belongs to God and must obey God.
The very reason we obey the State is that God established it; yet in that very affirmation, we also see that the State must obey God, and that inasmuch as the State departs from the law of God, it no longer deserves our allegiance. The apostles faced this reality when they were ordered not to speak again in the name of Jesus. "We must obey God rather than men," was their response (Acts 5:29). We admit the same principle in our own "Pledge of Allegiance" when we declare that this is "One nation, under God, with liberty and justice for all." That is what we are pledging allegiance to. If the nation is no longer under God, or no longer secures liberty and justice for its people, neither does it deserve our allegiance.
We can all recognize a legitimate separation of Church and State. The State, for example, cannot decree that we have six sacraments instead of seven. Nor can the Church decree that we have 51 states instead of 50. But if a new religion came along which had, as part of its worship service, the torture and death of infants, should the State step in and prohibit that? Certainly, such a prohibition would not violate the separation of Church and State. The reason is simple: such separation can never justify violence. That is precisely why the Church's active stand against abortion is not a "meddling in politics" or an "imposition of belief." Neither the Church nor the State can be passive if the other transgresses fundamental human rights.
The Church provides a necessary safeguard against the danger that the State becomes totalitarian, and that whatever human power decrees becomes, by that very fact, right and good.