I have been bothered by this question, and began looking into it more seriously lately.  I had assumed from polls and conversations with Fundamentalist Christianists who seem to have inordinate influence over the issue, that mainstream Christians were AWOL in the debate.  I did discover that recently many Christian churches are starting to work for health care now, though I don't know what their standards are.  AMike has links to some of these groups.

It is also true that many American Mega-churches have strong political arms and tons of money to exert their influence in politics, whereas more mainstream churches don't.  The political aims and wishes of the many who would like to see the United States ruled by Biblical law are another and larger issue.  But why, if they are heeding Christ's message and abundant example, would they not support Universal health care? 

The main doctrinal justification seems to lie within the Prosperity Doctrine, or The Word of Faith, which was originated by Ethan O. Allen in 1846.  He expounded on the idea that true believers were promised both healing and prosperity as a right.  It was the beginning of the modern Pentecostal movement, and spread to others like Phineas Quimby who taught that sickness was just about Wrong Beliefs, that man is made in God's image, God is not about sickness, ergo, if you are Godly, you cannot be sick.  Quimby's followers spread the teachings into New England, and it developed into the school of New Thought,  which taught that Confession and the Spoken Word (if Godly) would accomplish healing through Confession,  or speaking out loud the correct Scripture.  Mary Baker Eddy's Christian Scientist ministry stemmed from these teachings.

They also believed that God and Jesus promised prosperity, though the Biblical evidence for this is weak at best.  They relied on allegedly Old Testament verses (Old Covenant), and seem to blithely ignore most of Christ's actual teachings (New Covenant) concerning seeking and accumulating wealth.  They rely on a few quotes about "eventual riches" as meaning actual wealth, as opposed to spiritual wealth, a fact which critics attack vociferously.  Benny Hinn:  "Jesus rode a donkey, which in modern times would be equal to a Cadillac."  "Abraham and his friends were all wealthy."  "Jesus had a treasurer."  (Jesus now would have be called a Socialist, really.)

As television ministries grew through the work of Kenneth Copeland, Creflo Dollar, Benny Hinn, Joyce Meyer, Joel Osteen, Pat Robertson and others, the doctrine spread around the world.  It was a neat idea:  If you speak the Word of God, you will be healthy and rich.  If you send money to Our Ministry, you will be doing His Work.  You will get returns on your donations a hundredfold, a thousand-fold, whatever.

 Dollars flocked into the ministries, allowing the Mega-church ministers to flaunt their riches: fancy cars, huge church campuses, private jets, fleets of cars, expensive (though usually tasteless) clothes.  Then came newer and larger churches, increased teevee coverage world-wide, more money coming into the ministries, around and around.  And it's all been done within tax-exempt status for the Ministries, even though many churches support political candidates to advance their aims.

The Health and Wealth theology has plenty of critics, of course, who say the Bible repeatedly warms against the pursuit of riches, and deem Word of Faith ministries "cults."  Critic John Piper said in a sermon in 2006, "The prosperity gospel will not make anybody praise Jesus; it will make them praise prosperity."

From Worldview Interactive, Australia:

Some verses that you are unlikely to hear prosperity teachers use:

 'Seek first the Kingdom of God and His righteousness...'3
'Keep your lives free from the love of money and be content with what you have...'4

'But if we have food and clothing, we will be content with that. People who want to get rich fall into temptation and a trap and into many foolish and harmful desires that plunge men into ruin and destruction. For the love of money is a root of all kinds of evil. Some people, eager for money, have wandered from the faith and have pierced themselves with many griefs. But you, man of God, flee all this...' 5

'Give me neither poverty nor riches, but give me only my daily bread. Otherwise I may have too much and disown you and say, 'Who is the Lord?' Or, I may become poor and steal, and so dishonour the name of my God'6

'Though your riches increase do not set your heart on them.'7

'Whoever trusts in his riches will fall but the righteous will thrive like a green leaf.'8

'Cast but a glance at riches and they are gone, for they will surely sprout wings and fly off to the sky like an eagle'9

'It is hard for a rich man to enter the kingdom of heaven. Again I tell you, it is easier for a camel to go through the eye of a needle than for a rich man to enter the kingdom of God'10

'You will be made rich in every way so that you can be generous on every occasion...'11

'Command them [the rich] to do good, to be rich in good deeds, and to be generous and willing to share...'12

'Woe to you who are rich for you have already received your comfort'13

'Now listen you rich people, weep and wail because of the misery that is coming upon you.'14

'You say, 'I am rich, I have acquired wealth and do not need a thing.' But you do not realise that you are wretched, pitiful, poor, blind and naked.'15

Scripture References:
1. Acts 5:1-11; 2. Luke 12:33; 3. Matt 6:33; 4. Heb 13:5 5. 1 Tim 6:8-11; 6. Proverbs 30:8,9; 7. Ps 62:10 8. Prov 11:28; 9. Prov 23:5; 10. Matt 19:23,24; 11. 2 Cor 9:11; 12. Luke 6:24; 13. James 5:1; 14. 1 Tim 6:1

Ministries like Copeland ministries and Pat Robertson's programs actually weigh in on investment advice; finance is that much a part of their sort of Christianity. 

The many millions of Born Again Christians seem to find wealth virtuous, and believe their Hero Ronald Reagan's stated belief that it is important that there are wealthy Americans; they provide jobs for the rest of us.  Plus: if you pronounce the Word of God, you will be prosperous and healthy.  We tithe to our churches, not the government, which we all know is the problem, not the solution.  The message, worldwide, is enticing to the very poorest people, who would just love to find a way out of poverty.  Oprah's darling author of "The Secret, " Rhonda Byrne, had much the same message: You can visualize your way to prosperity; Shirley MacLaine professes that:  the poor are just blocking spiritually.  I don't mind these beliefs per se; but the point at which the Mega-ministries mega-money inject themselves into policy debate, I have issues.  Their money helps fund groups like Freedom Works, Americans for Progress, anti-cap-and-trade groups, anti-EFCA groups, etc.  It's curiously bent thinking or belief to my mind, and seems to run counter to Christ's words. 

One avowed "spokesman for Christians" I saw on teevee was asked by the host, "If Christ returned today, what do you think his opinion would be on your political acivities?"  She howled with laughter and said, "He'd be here, getting us all ready for The Rapture; He wouldn't be make pronouncements about all this stuff!"

(I can't find a link to Amike's blog.  Can you provide it, Amike?)