If the Republican Party wants to survive in the long term, it should change with the times on cultural issues and preserve the elements of conservatism that will make it strong. Despite the panicked fears of pundits and talk-show hosts, the United States is not moving in the direction of socialism and more government control of the economy--in fact, it has been moving in the opposite direction since the 1960s. The core Republican beliefs in individual economic freedom, individual responsibility, and the right to competitive self-betterment are not going out of vogue in this country, but the cultural conservatism of the religious right is--on same-sex marriage, abortion, and secularism especially.
The Moral Majority heritage of the 80s notwithstanding, the Republican Party has a strong core of economically conservative, civil libertarians who are becoming more and more turned off by the party's rhetoric on cultural issues. There are also many independents who can be drawn into the GOP's big tent if they are accommodated. While adapting their platform on cultural issues might prove unpopular in the short term, the Republican Party can avoid long-term extinction by doing so. Otherwise, the party is likely to fracture like the Democratic Party did in the late 1960's.
I am a hardcore liberal, and will be until the day I die. But even I can recognize that some Republican leaders such as Eisenhower and Reagan held great influence over the nation, and that they were loved by many. The Republican platform has shifted greatly, and as a result, it's followers are falling off.
Social issues such as reproductive rights, torture, and marriage equality have become headlines on Fox News. Political pundits such as Ann Coulter and Rush Limbaugh bombard listeners with right-wing propoganda that is so extreme, some are turned off. They preach of forcefully converting other countries to Christianity, women as inferior beings, and Bush as a figurehead. Some moderate Republicans may listen to this and become enraged at the bad image of the Republican party being displayed.
In addition, the party is losing entire classes of voters such as women and LGBT voters, as they continue to insult these groups with their continuous talk of social issues that are important to these people. As a general rule, offending potential voters is not a good way to win an election.
Recently, a rise has been seen in the conservative support of marriage equality. The train is moving slowly, but at least it's moving. A recent poll suggests that to win the election in 2016, the candidate running (regardless of which party) must publicly support marriage equality or not oppose it to win the election. This is just another indicator of the American people speaking to the right-wing extremists and even the Bible-toting moral preachers opposing something that is going to happen.
Finally, the ties between the Republican Party and the Christian Church HAVE to become less dominant. Fundamentalist Christians may be chanting "Yes! Yes! Praise the Lord!" as they watch Fox News with a Bible in hand, but the rest of the nation isn't as emphatic. Non-Christian voters may be turned off by the exlusivity "Bible or Bust" paradigm of the conservatives.
Both parties should get out of the censorship business. Government was not meant to protect us from cultures which some might find offensive, but from war, domestic threats and the likes. In our country we are given rights by the Constitution such as the right to free speech, the right to assemble, the right to bear arms, however one of these rights is not the right to not be offended. Their is no need to protect popular speech, so why then would we have the right to free speech (speech extends to any form of expression)?
To me, it seems that the basic liberal and democratic approach to culture is to let it evolve itself without trying to regulate it at all unless someone's culture is violating someone else's rights. However, it would seem that the conservative/republican mindset is to regulate other people's culture and morals based on THEIR own culture and morals, which is just fundamentally wrong.
The culture war is the the republicans pushing for their believe system or set of believes. So what you are saying when you ask "Should Republican give up their fight in the culture war?" is "should they stop fighting for their believes so I can freely exercise mine." Of course they should not give up their fight in the culture war. What kind of a question was that to even ask? Not A very well thought out one.
I believe in the preservation of our culture and our values. That's what a nation is, it's a union of our collective ideals and values, which then make up our culture, our culture is our nation, and it should be protected and preserved as such. While this does not give an excuse for staying in the past, culture should be preserved, and progressives should be countered with action that will tell them that we will not let them manhandle our culture no longer for their personal desires and letting immoral degenerates have control over it.