Whether we should allow our faith to affect our political decisions is irrelevant. everyone's political outlook is affected by their worldview. So from that perspective, it's a question based on a false assumption that we can separate our beliefs from how we vote. Everyone does that.
As for whether Christians should "force their views" on others, that's also flawed for the same reason. The point of voting is to vote your views. Atheists, Muslims, Wiccan, doesn't matter. Everyone votes their conscience.
Fundamentalist young earth creationist Christians , for example, feel that outlawing any mention of the idea that there is a Creator is the atheists "pushing their views" on people. Atheists, on the other hand, feel that YEC groups trying to get Creationism included in the classroom constitutes the YEC Christians trying to "push their views on others." Who is right and wrong is a matter of debate and neither side is likely to agree with the other. Ever.
In the United States, as a Constitutional Republic, it is expected for people to vote their conscience. That's why we have freedom of religion (not from, of), freedom of (not from) speech, freedom of assembly, etc. But that's a political question, not a Christian one.
As for an angle that includes a denominational or historical Christian view:
This was the concern of the Danbury Baptists, who were concerned that the government would impose laws governing religion. They believed that the protection of religious liberty was important, and it is based on their concern that Thomas Jefferson coined the phrase "separation of Church and State". It was meant to protect individuals from governmental interference, not the other way around.
The Danbury Baptist Association of Danbury, Connecticut sent a letter,
dated October 7, 1801, to the newly elected President Thomas
Jefferson, expressing concern over the lack in their state
constitution of explicit protection of religious liberty, and against
a government establishment of religion.
In their letter to the President, the Danbury Baptists affirmed that
"Our Sentiments are uniformly on the side of Religious Liberty — That
Religion is at all times and places a matter between God and
individuals — That no man ought to suffer in name, person, or effects
on account of his religious Opinions — That the legitimate Power of
civil government extends no further than to punish the man who works
ill to his neighbor..."
As a religious minority in Connecticut, the Danbury Baptists were
concerned that a religious majority might "reproach their chief
Magistrate... because he will not, dare not assume the prerogatives of
Jehovah and make Laws to govern the Kingdom of Christ," thus
establishing a state religion at the cost of the liberties of
Thomas Jefferson's response, dated January 1, 1802, concurs with the
Danbury Baptists' views on religious liberty, and the accompanying
separation of civil government from concerns of religious doctrine and
practice. Jefferson writes: "...I contemplate with sovereign reverence
that act of the whole American people which declared that their
legislature should 'make no law respecting an establishment of
religion, or prohibiting the free exercise thereof,' thus building a
wall of separation between Church & State." This doctrine, known as
the "wall of separation" or "strict separationism," would later become
highly influential in 20th century Supreme Court understandings of the
relationship between church and state. As a result, the relevance of
this letter is a subject of heated debate, with scholars such as
Robert Boston emphasizing its importance, and others such as Mark
David Hall arguing that the letter was a historical outlier.