Let's assume Congress doesn't raise the debt ceiling. As you said, the Fed could just print the money. What would be the consequences of that? It seems to me there would be no negative consequences and a number of positive consequences.Here's a follow-up I posted.
The most significant potential negative consequence is that the world would lose faith in the Dollar as a stable currency and would treat it as one likely to face run-away inflation. But the administration could combat that threat by stating that once the debt limit is raised, we would sop up the extra money by borrowing it back. I think the administration and the Fed (although not the Congress) have the credibility to make a promise of that sort.
The positive consequence would be additional stimulus. We would be injecting lots of money into the economy without having to borrow it. That would increase the supply of money,which would lead to increased demand, increased asset values, and even lower (long term) interest rates.
So why shouldn't Obama say that if Congress doesn't raise the debt ceiling, the administration will simply print the necessary money until Congress acts?
Of course this whole issue is silly. As you said, there is probably no other country in the world with a debt ceiling. But as long as we're being silly, why not go all the way!
In effect, the government would be (implicitly) borrowing money from the Fed, which it would pay back when the debt ceiling is raised. To reduce the inflationary effect, the Fed could, in turn, sell items on its balance sheet into the market. By selling the same amount that it "lent" to the government, it would, in effect, be providing a way for the government to borrow without actually issuing new debt, i.e., it would be providing a debt laundering service for the government. This could go on until the Fed ran down its balance sheet—which would take quite a while.