Note that given the historical existence of the Papal States, the claim is, even when interpreted in the narrow sense, still wrong.
I think Sam is generally correct about Western Christianity. Yes, historically many church officials have simultaneously held secular power, and this was by no means limited to the Papal States -- as the most notable examples, in the Holy Roman Empire there were a great many sovereign abbeys and prince-bishops. These were theoretically under imperial authority, but in practice, except for the occasional appearance of strong medieval emperors, they were fully sovereign for all practical purposes. (Interestingly, besides the Vatican, another historical relic nowadays is Andorra, whose co-sovereign, in an odd arrangement, is the bishop of Urgell in Catalonia.)
However, in all these cases, the simultaneous secular and ecclesiastical authority was considered as two separate functions exercised by the same person or institution, not as one and the same. The closest modern analogy would be when a bishop as an individual, or an abbey as a corporation, is also the owner of some business enterprise. (Indeed, before the rise of the modern nation-state, the line between sovereignty and private property was much less clear.) The claim to the secular authority over a piece of land could be gained or lost by conquest or a legal transfer independently of the ecclesiastical title, even if they were often held and passed together for many generations. The secular issues would be under the jurisdiction of the local secular legal system, typically derived from some mix of the local customary law, Roman law, and sovereign statue, while the ecclesiastical issues would fall under the canon law.
This is very different from a real theocracy, where religious leaders claim secular authority by virtue of their religious status alone, and where religion is considered as the sole, or at least primary, source of law -- something that was never true for any legal system under Western Christianity. Even in the Papal States, the religious office of the Holy See and the sovereign office of the monarch of the Papal States were considered as separate. (And foreign powers could deal with either of these separately.)
This is not to say that popes never claimed theocratic secular authority. However, such claims were typically unsuccessful, and even their occasional temporary and limited success would backfire badly. Insofar as real theocracies ever sprang up under Western Christianity, it happened among peculiar fringe groups, such as the New England Puritans.
You also say:
There's a reason that the Catholic Church in the 19th century labeled the idea of separation of Church and State a heresy dubbed "Americanism".
However, the heresy of Americanism was about principled opposition to having an established church. But an established church is very different from theocracy. The fact that a particular church is awarded particular legal privileges, or even some kind of special representation in the government, doesn't mean that religious leaders are claiming secular sovereignty by virtue of their religious position, or that religious law is being enforced. England, for example, has had an established church throughout its history (there are still bishops in the House of Lords), and it was never a theocracy in any meaningful sense of the term.
Similarly, the idea that secular law should incorporate values dictated by the predominant or established religion is not theocratic either, as long as the law originates from legislators whose office is not religious, and doesn't claim a religious basis for its authority. Of course, a principled secularist may consider any legal privileges awarded to religious groups and any religious influences on the law as unacceptable. But claiming that any such things automatically constitute theocracy is an abuse of the term.
Also:
The Ottoman Empire for example did not have clergy members involved in politics (although technically speaking the sultan was officially considered to be the heir of the caliphate, I think.).
Ironically, for Christians living under the Ottomans, the millet system) forced a much greater blending of religious and secular authority than anywhere in the West. It is in fact one of the main reasons why I had to make the qualification about "Western" Christianity in the above paragraphs. (Though not the only one, as things were somewhat different in Eastern Christianity even before the Turks.)
That aside, the Ottomans were indeed a peculiar period in the Islamic history, and their rule was long and widespread enough that these peculiarities should be taken into account when making general statements about the Islamic world. On the other hand, it is undeniable that the basic theocratic notions, such as law derived straight from religious authority and lack of clear de jure separation between religious and secular offices, have indeed been much more characteristic of Islam than of Christianity -- especially Western Christianity, whose historical tradition of such separation is in fact the root of the modern notions about the separation of church and state.
England, for example, has had an established church throughout its history (there are still bishops in the House of Lords), and it was never a theocracy in any meaningful sense of the term.
Theocracy has a lot of meanings, and one can argue that Elizabethan England was a theocracy in that the head of state was the head of the religion, and that the official religion was compulsory, but I was contrasting Islam and Christianity, and Elizabethan England was not a theocracy in the sense that Islam is a theocracy, one difference being that law in islam suppos...
I wanted to bring attention to two posts from Razib Khan's Discover magazine gene expression blog (some of you may have been readers of the still active original gnxp) on the polemic surrounding Pinker's The Better Angels of Our Nature.
Relative Angels and absolute Demons (and the related But peace does reign! )
I generally agree with some of his arguments, but found this quote especially as summing up some of my own sentiments: