Do Christians have the right to govern? We've already seen that they do not. We've also seen that governing other people is immoral unless they agree to participate, and that restricting people's liberty like this is bad for the prosperity of society, because it's bad economics. (Basically it's socialism, even if you like to call it "democracy" and pretend it is "liberty.")
But people have the right to establish whatever voluntary associations they want. The members of those associations may delegate their authority to those associations. So, they may give or loan some of their property to the group. They may empower the group to act in their defense. They may not legitimately have the group take what doesn't belong to its members, or use force against non-members for purposes other than defense of rights. (If they do, they have founded a criminal organization, like the Mafia, even though (like the Mafia) the group may go around telling everybody that it is "protecting them" and so they need to "pay up," calling itself "the state" to pretend it is something other than a criminal organization.)
As a Christian, I take Jesus Christ as my King. (Contrary to popular premillenial theology, the Bible teaches that Jesus is sitting on the throne now and that Christians are members of His kingdom in the present tense, not waiting for it in the future. See Colossians 1:13.) With the Lord as my King, I submit to His commandments, and His administration. We won't study all that the Bible has to say about church organization here, nor cover the differences of opinion within different churches on the subject, but the Bible indicates Jesus commands that I be part of a church or congregation, and subject to the leadership of that congregation, which from my reading I understand to be a group of bishops. The exact organization of your church may vary somewhat from mine, and I'm not going to debate that here, but either way, we are each part of what is essentially a voluntary association of human beings. We are all of us members of our churches voluntarily; for the most part, nobody is a member of a particular church because they were told to be. And to some extent, we are governed by our churches.
Rather than defending my rights, the leadership of my church is empowered to "watch for my soul" (Hebrews 13:17). They have a handful of tools, a set of authorizations from the Lord, in order to accomplish this. The primary method is by teaching. But if I should start having problems in my spiritual walk, they are then supposed to rebuke me. However, they are not permitted to use force against me. Their job is persuasion, not compulsion. They aren't told they can imprison me or fine me (Acts 5:4) to try to get me to do what is right. At the last, if I refuse to repent, they are empowered to put me out of the church. This process is described in I Corinthians 5:5 as "delivering such a one to Satan for the destruction of his flesh, that his spirit may be saved in the day of the Lord Jesus." Once a person has been put out, he is treated "as a heathen and a tax collector" (Matthew 18:17).
Outside of the church, the church has no power whatsoever to judge. This is explained in the rest of I Corinthians 5, which purposely specifies that this limited judging power of the church is only employed against believers:
9 I wrote you in my letter not to associate with immoral people;
10 I did not at all mean with the immoral people of this world, or with the covetous and swindlers, or with idolaters; for then you would have to go out of the world.
11 But actually, I wrote to you not to associate with any so-called brother if he should be an immoral person, or covetous, or an idolater, or a reviler, or a drunkard, or a swindler-- not even to eat with such a one.
12 For what have I to do with judging outsiders? Do you not judge those who are within the church?
13 But those who are outside, God judges. Remove the wicked man from among yourselves.
So the Bible is very clear: Christians are governed by their churches, but not permitted to judge those outside. Within the church, you may confront someone about their immorality, teach them, persuade them, rebuke them, and encourage them to repent. But if you are passing laws that punish immorality for people outside of the church, you are in violation of I Corinthians 5. You are judging outsiders. God judges those people. Leave them alone.