The now-deleted writer of that post mentions Res Publica Christiana which frankly is something of a post-facto historiographic idea. It has a problem on two fronts. The first is that the self-conception as a Christian ruler is utterly obvious: no ruler would conceive of themselves otherwise; the 'binding' nature of this is a shallow analysis and doesn't help us understand medieval rulership. The second problem is "went back to the Code of Justinian of 529". This is total hokum: not many souls in western Europe knew anything about Justinian Code prior to around 1050: Justinian's Codex was an abortive effort in the Emperor's own time and had relatively no impact for centuries even within the Byzantine Empire.

Both problems above are the products of historians who believe that the way medieval rulers and their chancellories wrote about their decisions and beliefs were direct reflections of the world, and not polemical and strategic positioning. It's a source selection bias.

Clearing away the crap, we come to the question again. As I said above, 'Christendom' as some 'unifying concept' is really a shallow way of looking at medieval governance. It's utterly obvious and provides no explanatory power against the various interpretations of power by medieval rulers and their minsters. The best we can say is that 'Christian' was a polemical and ideological tool; this isn't to say that rulers and their ministers weren't devout Christians. But it is to say it's not enough of an explanation of how governance operated.

When we come to discussing 'legal systems' or 'doctrines' we then must be very specific about epochs within medieval history. There is a fairly sharp dividing line between pre- and post-12th century western Europe. We cannot say there was a 'judicial system' of 'legal system' prior to the 12th century, not even within Carolingian society - there was no centralized 'judicial process' in early medieval Europe - it was driven by councils making decisions based on certain capitularies. But the form of those councils (placitum) and lower-level 'trials' were entirely customary. Even in the 13th century it was still taking shape. I wrote about one aspect of this shape-taking a few weeks ago here.

Now, if we speak about 'legal systems', we need to separate two parts at this point. The first part is ideological: how did rulers, their ideologues, and the public conceive of rulership after say 1200? This is a subject that perplexes historians to this day, and I can't address it here in any way that is useful.

However, the other part is about 'systems'. In my post I link to above I write at length about the development of some judicial processes in the 12-13th century. In fact, we can speak here of centralization, of systemization of legal processes; the opposite of the 'fragmented' systems of the early medieval period. Fragmented here must be explained: the word itself suggests some need for cohesion and consistency, some over-arching ethic. It would be wrong to think that prior to the 12th century western Europe was without law, without consistency. However, that law was customary and because it was customary it was subject to local opinion and patterns of use. It was not, however, bound by consistency of process across jurisdictions. The innovation of the scholars at Bologna, and of the Papacy, in 12th century was the resurrection of judicial processes, ordo iudiciario. This rapidly became 'internationalized' (that word is anachronistic for this period). This was the basis of the transformation of European legal systems, including common law systems of England (and thus the colonies like US and Canada).

The 12th century origins of our modern justice systems were in the Church and Canon Law; and it continued as such through the following 500 years. Canon and secular jurisprudence were tied together in universities, undivisible; this was called the ius commune. It wasn't until the nationalization of law in the 18th century that the ius commune was ruptured, and that local law ('customary law') was subordinated to national law.

From the 12th century through the end of the middle ages and on into the early modern period, customary law still ruled to a great extent. Although its rule was bound by judicial processes that in fact created European legal processes (and a 'system') as of the 13th century.

By way of example, France in the late medieval period had 'parliaments', or King's Courts, in 13 locations. These courts dealt with inter-nobility appeals. But step down below high nobility and you were dealing with variable, local customary law, even if it was bound up in 'modernized' judicial processes (of evidence, witnesses, roles of appellant and defence).

To be perfectly honest, I believe any attempt to create an idea of 'international' legal relations anytime before the early modern period is anachronistic. Kings, lords, bishops, abbots, the Pope and his legates, all fought battles for power, most of which had little to do with 'law' so much as hubristic appeals to God, amongst other ideas. There was no international court of the Hague to fight these things; these things were dealt with on battlefields, in letters and missives, and sometimes in face-to-face meetings. But the idea of Christian as 'unifying' among rulers was only as good as the other party's willingness to concede a point; it was not for nothing that rulers would accuse each other of heresy.