Does "freedom require[] religion"?

In his "Faith in America" speech, Gov. Mitt Romney said, among other things:

Freedom requires religion just as religion requires freedom. Freedom opens the windows of the soul so that man can discover his most profound beliefs and commune with God. Freedom and religion endure together, or perish alone.

Is this true?  Over at Balkinization, Jack Balkin says that this statement (and some others) in the speech "strongly identify Americans and Americanism with belief in God."  Is this true?

For starters, it is (obviously) not the case that only those persons who believe in God (or who, in Gov. Romney's words, "believe that Jesus Christ is the Son of God and the Savior of mankind") are or can be good Americans.  It is certainly not the case that only such persons desire, deserve, and sacrifice for "freedom"; or that only a political community consisting primarily of such persons can be "free."

I do not know exactly what Gov. Romney intended to communicate or claim with the statement that "[f]reedom requires religion".  If he intended to claim with that statement what (it sounds like) Prof. Balkin understands him to have claimed, he was mistaken.  (It seems to me unlikely, though, that Gov. Romney believes that only theists -- or Mormons, for that matter -- are or can be good Americans.)

That said, I believe that it is true -- or, at least, that there is a sense in which it is true -- that political "freedom requires religion."  To be clear:  It is not true that a political community of religious people will, necessarily, be "free"; or that a political community in which most people do not believe in God cannot be "free"; or that religious believers will always cherish, protect, respect, or even understand political freedom.  (I assume that Prof. Balkin and I agree entirely about all this.)

All that said, it seems to me that the existence and maintenance of political freedom does depend on -- i.e., does "require[]" -- "religion" in the sense that political freedom requires not merely constitutional or other legal limits on government power and official action, but also that (and a consensus that) the aims, sphere, authority, purpose, reach, and nature of the state -- of politics -- be limited, by something else.  That is, it is crucial to political freedom that -- in Harold Berman's words -- it not be "for the secular authority alone to decide where its boundaries should be fixed" and that -- as John Courtney Murray put it -- there be "room for the independent exercise of an authority which is not that of the state."  And, it seems to me that "religion" is best, and perhaps only, able to satisfy (even though, of course, it has often failed badly to satisfy) these requirements.  (I tried to flesh out this idea in this short paper.)

What do others think?  Is this, or something like this, plausible? 

I cannot emphasize this enough:  To suggest this is not, at all, to say that only religious people understand the value of, and cherish, freedom-under-and-through-limited-government.  None of this is intended to be -- or, objectively, is -- exclusionary, triumphalist, "theocratic", or "Christianist."  Nor am I claiming that these thoughts of mine capture or reflect what Gov. Romney intended to say.  I do not know, exactly, what he indended to say.  (Disclosure:  I am a member the Thompson campaign's Law Professors Committee.)