"When Gentiles, who do not have the law, by nature, do what the law requires, they are a law to themselves, even though they do not have the law" (cf. Rom. 2:15, ESV).
We have been given the Great Commission and the Great Commandment, but is there a Great Cultural Mandate? Some refer to a social gospel, but this is a misnomer. Our ultimate allegiance is to Jesus, naturally, not to the government, and Jesus did say that His kingdom is not of this world--that He was no threat to Caesar. His kingdom is not of this world! What kind of social calling do we conclude from Scripture? Some Christians believe this society isn't worth saving, some believe politics can't change the human condition, some believe the Bible is not relevant to civil government, and some do believe we should secure and take back society for Christ or reclaim America for Christ! Is it our calling to save America?
You can get too involved in the politics of this world that you are getting nowhere spiritually and getting your eyes off Jesus, but there are evils in society that must be rooted out and the government is instituted to overcome evil, restrain it, and contain it. Look at some of the greatest social movements in history, such as ending the gladiator fights, ending slavery, ending child labor, and giving women their rights, are the result of Christian influence--so there is a place for Christians to get involved with a divine calling.
We also need Christian influence in the government, i.e., as elected officials and public servants influencing for the good as salt and light. Christ's kingdom is one of the hearts of men and is not visible, it will be set up visibly at His second advent and we will rule with Him in glory. But some Christians believe we should try to take over and impose our way of life on the infidel: making him behave or live in a Christian manner and standard. Christian law has been tried and failed several times in history: Calvin's Geneva; Bradford's Plymouth; Winthrop's Massachusetts' Bay Colony; Cromwell's England, among others. We frown upon Muslims saying that tomorrow belongs to Islam and when they set up Shariah law, they aren't recognizing the rights of minorities, i.e., Christians. How can we expect non-Christians to live like Christians? This is really Christian Shariah law (like blue laws, gambling restrictions, and prohibition).
The purpose of government is not to outlaw sin but basically to protect freedoms, protect property, punish crime, securing justice. We are to "render to Caesar that which is Caesar's" (cf. Matt. 22:21) for sure, and we are never told to eschew society, start a revolution, or join hands with the powers that be as partners in crime either. We don't have carte blanche to wreak havoc on the social order. If we lived in a communist nation, we are not to become involved in a subversive underground of resistance but be model citizens and fulfill our mission of being salt and light.
However, there does come a time when the government tells us to do something immoral or forbids the free exercise of our faith, then it's imperative to disobey in an act of defiance and civil disobedience. Here's the rub: "We must obey God rather than men" (cf. Acts 5:29; 4:19). This is amply demonstrated by Daniel refusing to pray to the statue and being thrown into the lion's den; he neither flaunted nor privatized his faith as a prime minister who was salt and light.
The Bible makes it clear that our primary mission is to change hearts through the gospel message, and hopefully, society will be revolutionized as a direct result. Paul didn't make it his mission to overthrow slavery, but with the spread of Christianity, its demise was certain as believers rejected it and had more influence--not forced change. We are not called to set up a theocratic government or experiment with a nation ruled by religious leaders. It was Christian influence that said we have the right to life, liberty, and the pursuit of happiness--rights are conferred from God and recognized by the state.
However, the conclusion of the matter is that we represent Christ to the world as His hands, feet, heart, and voice and we have the power to make a difference by being that salt and light, which preserves and shows the way in the darkness. The church shouldn't get overly obsessed with turning stones into bread or mission work unless the gospel is propagated with it, we are not to become mere do-gooders and must remember and not lose focus of the fact that we are "aliens and foreigners" and our real "citizenship is in heaven" (cf. Phil. 3:20). Christians do have a duty and responsibility to the state, though, as Edmund Burke phrased it: "All that is necessary for evil to triumph is for good men to do nothing."
Soli Deo Gloria!