People whose values were derived from their religious beliefs have just as much a right to exercise their political rights as do those who formed their values from classes in school, secularism, their parents, the Green party, rock music, movies, etc. It is not legitimate to say that anyone who goes to a church or synagogue has no ability to think and should be denied rights every other citizen has.
Further, some matters are obviously right or wrong even without referring to a religious dogma. Although I am a Christian, I don't look at the Terri Schiavo case as a particularly religious issue. I think of it as a judicial problem. The judicial system allowed a man of questionable intent to decide whether a disabled woman would be given food. I think that Judge Greer made a woefully bad decision. Where there is doubt as to her desires and her ability to function, the default should have been to let her live, not to kill her. The rest of the judges followed along and went by the letter of the law rather than the spirit. You don't have to be "religious" to find fault in their decision.
There are basic moral issues that the Christian conservatives are more vocal about. However these moral values are not the sole provence on Christians. The value of life is enshrined in Christian theology, However it also valued among other religions, such as Judaism and Buddhist and Hindu. I am not an expert on religion. Even atheists or secularists can agree on the value of life. Some put more emphasis on the quality of life. Christins are not the only ones that think gay marriage and abortion are wrong.
I don't think that the Republican Party is hostage to the Christian right. It is only that there are shared basic values.
Values are incorporated in law and religion. Early law was done through religion. Islamic Sharia law is an example of this. I don't sgree with Sharia law but can understand that it has greater force if posited through religious dogma.
Without basic shared moral values civilizations can not exist. Thou shall not kill, steal, defame etc. are necessary for people to reside together. To fight to maintain those values is a fight for survival of our society. Since we are a Republic, when the people see that we are straying from these core values they are correct to agitate for change or to prevent perversion and degeneracy from becoming the norm.