Gaudium et Spes 76 treats a bit of the Church-State issue, starting with the need for people to know the separation between the two in mission:
It is very important, especially where a pluralistic society prevails, that there be a correct notion of the relationship between the political community and the Church, and a clear distinction between the tasks which Christians undertake, individually or as a group, on their own responsibility as citizens guided by the dictates of a Christian conscience, and the activities which, in union with their pastors, they carry out in the name of the Church.
The Church stands outside the realm of politics, holding up the ideal of the free and noble character of the human person, as intended by God:
The Church, by reason of her role and competence, is not identified in any way with the political community nor bound to any political system. She is at once a sign and a safeguard of the transcendent character of the human person.
Though separate in purpose and vocation, there are mutual goals shared:
The Church and the political community in their own fields are autonomous and independent from each other. Yet both, under different titles, are devoted to the personal and social vocation of the same (people). The more that both foster sounder cooperation between themselves with due consideration for the circumstances of time and place, the more effective will their service be exercised for the good of all.
The council bishops outline the goals of the Church, and remind readers that the People of God transcend nations and their particular values, shared and unshared. The Gospel is also meant to be applied in “all fields of human endeavor.” Doesn’t sound very weak to me:
For (human) horizons are not limited only to the temporal order; while living in the context of human history, (a person) preserves intact (an) eternal vocation. The Church, for her part, founded on the love of the Redeemer, contributes toward the reign of justice and charity within the borders of a nation and between nations. By preaching the truths of the Gospel, and bringing to bear on all fields of human endeavor the light of her doctrine and of a Christian witness, she respects and fosters the political freedom and responsibility of citizens.
The Apostles, their successors and those who cooperate with them, are sent to announce to mankind Christ, the Savior. Their apostolate is based on the power of God, Who very often shows forth the strength of the Gospel on the weakness of its witnesses. All those dedicated to the ministry of God’s Word must use the ways and means proper to the Gospel which in a great many respects differ from the means proper to the earthly city.
This chapter concludes by affirming the Church’s freedom to teach and to judge situations in the secular sphere:
There are, indeed, close links between earthly things and those elements of man’s condition which transcend the world. The Church herself makes use of temporal things insofar as her own mission requires it. She, for her part, does not place her trust in the privileges offered by civil authority. She will even give up the exercise of certain rights which have been legitimately acquired, if it becomes clear that their use will cast doubt on the sincerity of her witness or that new ways of life demand new methods. It is only right, however, that at all times and in all places, the Church should have true freedom to preach the faith, to teach her social doctrine, to exercise her role freely among (people), and also to pass moral judgment in those matters which regard public order when the fundamental rights of a person or the salvation of souls require it. In this, she should make use of all the means-but only those-which accord with the Gospel and which correspond to the general good according to the diversity oœ times and circumstances.
While faithfully adhering to the Gospel and fulfilling her mission to the world, the Church, whose duty it is to foster and elevate (Cf. Second Vatican Council, Dogmatic Constitution on the Church, n. 13: AAS 57 (1965), p. 17.) all that is found to be true, good and beautiful in the human community, strengthens peace among (people) for the glory of God.(Cf. Luke 2:14.)
There are times when Gaudium et Spes rambles or fails to make a point succinctly or adequately. But passages like this give lie to the argument the document fails to be strong enough in staking out Gospel ground.