(The following is not an apologetic for Christianity. If you are not a religious Christian, you may find some of my statements offensive. I will only be supporting any questions about what I have to say with Scripture, so you may find it unhelpful. Sorry for the inconvenience. An apologetic is best left for another time.)
Where the religious right and leftist or liberal Christians differ often regards what to do where society is in direct disobedience to God. All genuine Christians acknowledge that the world is in sin and it must be saved. This salvation can only be accomplished by God.
How do we react to sin in ourselves? How do we react to sin in our fellow believer? How do we react to sin in unbelievers? The answer to the first question dictates the answer to the other two.
All genuine Christians who have been converted by the power of the Holy Spirit know that we are called to flee from sin and temptation, put off the "old man," produce good spiritual fruits and strive towards obedience. When we sin, we are expected to repent and turn from our sin quickly. We are also expected to produce spiritual fruits worthy of repentance. (I Kings 8:46-48, Job 34: 32-33, Isaiah 59: 20, Jeremiah 15: 18-20, Jeremiah 31: 18-20, Matthew 3:8, Acts 2:37-39, Acts 20: 20,21, II Corinthians 7:10, 11 etc...)
Dealing with a fellow Christian who sins is no different. There are many portions of Scripture that instruct us how to do this. Matthew 18 states that when you see your brother in sin, you are to go to him directly and quietly and confront him with it. Leviticus 19: 17 states that we are to rebuke our brother frankly and so avoid sharing in guilt. It is important to notice that Christians are not given any license to ignore their brother's sin. There is nothing in the Bible that can be used to argue "what's good for you is good for you and what's good for me is good for me." Far from it. Nathan did not go to David and tell him it was ok to sleep with Bathsheba. Elijah did not tell Ahab and Jezebel they could continue to serve idols. Peter did not tell the crowds that they did right in executing Jesus. If we ignore the sin of our brother and do not rebuke him, then we share in the sin. We are disobeying God's Word if we do not rebuke our sinning brother, since there are so many verses instructing us to do it. (Leviticus 19: 17, Ezekiel 33:1-9, Matthew 18: 15-17, Luke 17:1-4, Galatians 6:1-3, I Timothy 5:19-20, I John 2:1, I John 5:16-18, etc...) Furthermore, we show the ultimate lack of love when we allow our brothers and sisters to walk a path we know could end in their ultimate destruction. Our obedience to God will naturally be reflected in how we deal with the sin of our fellow Christians: rebuke in humility and love and upon their repentance, forgive.
This practice is also extended to the world. The world is not to be left to itself. Christians have been given the Great Commission (Matthew 28: 18-20). We have been instructed to go into the world and preach the Gospel. Everyone is to repent and believe, be saved and be baptised. There are no exceptions. God demands worship. He demands obedience. We are to live, preach and teach His Word.
Put another, very unpopular way it's this: what God says is good for me, He says is good for you. Not only does He expect me not to sin, He expects you not to sin. He demands your obedience just as much as He demands mine. He is deserving of your worship just as He is deserving of mine. Anyone who tells you the Bible does not say this is either ignorant of what the Bible says or is deliberately lying to your face. Every knee will bow and every tongue will confess God (Isaiah 45:22-24). It's only a matter of whether you bow now in life or after you are dead.
Every Christian knows that this is the one aspect of our message that unbelievers do not like to hear. It can make for such an uncomfortable face to face conversation. It can result in arguments, broken friendships, insults being hurled and all kinds of accusations of being judgemental. As Christians we can either obey God in this matter, or we can sin and say nothing and make the world feel comfortable.
An important side note must be made before I continue. Christians may not preach to the world or rebuke their fellow believer and continue to walk in sin themselves. This is hypocrisy. Rather, our attitude must always be "but for God's grace there go I."
Back to politics.
The religious right proceed from the belief that God is deserving of everyone's worship. Worship is not restricted to Sunday morning church services. Worship encompasses the entire life of a Christian. God is to be first in everything. Our personal lives are to reflect Him. Our homes are to reflect Him. Our businesses and schools are to reflect Him. Our healthcare, laws and treatment of the poor are all to reflect Him. This is not the Church interfering with the State, lest anyone should think that argument could apply. This is the State desiring to do God's Will. The religious right does not argue that the State be compelled through force to do as the Bible commands. Instead, it argues that the State should desire to do God's will. It is simply another way of articulating to the world that it is lost and must be saved. If the State desired to do God's Will, then it would not pass laws sanctioning things like same-sex marriage.
Put another way, the religious right are doing what every Christian is supposed to do. The religious right call the world, specifically the State, to repent and be obedient to God. The only difference is that they use the political forum to do so. This is not forbidden in the Bible, but encouraged. This was the role of every prophet who ever lived: to remind the king Who it was that put him there.
Because as Jesus said to Pilate, you have no authority unless it is given to you by God.