In a word, no.
In fact, as this article illustrates, it would cost less than the current system, through reduced administrative costs and the ability to negotiate prices on a nationwide level. This is why single-payer exists and works well in every other industrialized nation in the world. But not in the United States, the richest country in the world.
This is just another example of how conservatives smear a perfectly good idea by calling it "socialism." The only socialistic aspect about it is that it would not allow private companies to profit off of the misery of others. Health care is not like other private goods and services: you never know exactly when you're going to need it. That's why it makes the most sense to spread the costs of health care out among the entire population. That, in fact, is the basic rationale behind all insurance schemes, even private ones.
Will we ever get past labels and just look at and discuss ideas? Sometimes, I wonder.