This is by no means my area of expertise buuuuuuut here goes nothing.
SHORT ANSWER: Like most generalizations about medieval history, it may be broadly true but overlooks change over time and specific nuances that in fact tell a fairly interesting story about the creation of certain institutions. The background for this is the end of the Roman legal system (which in its own way gave power to those with a legal education or the money to access one, to a level that might outstrip the power imbalance of medieval european law) and the creation of the Germanic legal systems (sometimes, as in the case of the Visigoths, with Roman holdovers) in the post-Roman West.
One of the major themes of medieval history (which has been treated at length in the historiography of the period) is the transition from what we might call personal or charismatic law to a system of law that we might see as more modern: nominally impartial, with rights of appeal, etc. This situation is complicated by the fact that the medieval period had many different types of law. Even more than differing national (in the modern, anachronistic sense) laws (e.g., the law of England vs France vs each little part of the Empire) there were overlapping spheres of law. In Visigothic Spain, for instance, there were different ethnic laws for Romans, Goths, Jews, although how this worked in practice is still up for debate. Similarly, in al-Andalus there were complicated rules about who could appeal to which courts in which cases. There were also conflicts between the Church law (canon law) and the secular laws of the lords, which often but not always expressed itself in the debate over what scholars today call "the papal monarchy."
With the caveats above--that there are overlapping systems of laws, and that things worked differently in different places, and at different times--we can make some broad arguments about the shift from personal law to institutionalized law. This is tied in with the question of the growth of the administrative state. The simplest argument is that as governments got more complicated and needed things like continuity and cohesiveness, all sorts of things started to happen--taxes needed to be collected, records needed to be kept, plans needed to be made--and all of these eventually created the conditions for centralization and standardization of power. At this time you also see the rebirth of a literate culture, what we sometimes refer to as the "documentary revolution." Furthermore, the "rediscovery" of Roman Law at the University of Bologna and its spread over the course of the twelfth and thirteenth centuries provided a model for even those places (like England) which did not adopt Roman law. By the time of the Fourth Lateran Council (1215) you had the idea that law needed to be universalized and rationalized, and thus you get the end of things like trial by ordeal.
It is in fact much more complicated than this. Part of this is because of the different levels of hierarchy. An interesting case is England, as described in Strayer's short but thought-provoking Medieval Origins of the Modern State. Strayer makes the point that the establishment of many of the things we take for granted in law today--jury trials, appeal, etc.-- came out of royal attempts to undercut the arbitrary power of local lords, and was tied up with the push for more efficient tax collection and the creation of the exchequer. So yes, while in this case there was a move away from what we might call arbitrary personal law (but really--customary law) administered by local lords, the centralization and rationalization (I hate that word) of the law actually served to increase the power of another lord--the King.
Here are some good classic sources: Bartlett, Robert. Trial by Fire and by Water
Bisson, Thomas. The Crisis of the Twelfth Century.
Cheyette, Frederick. “The Invention of the State.” In Essays on Medieval Civilization, ed. Bede Karl Lackner and Kenneth Roy Philp. Austin: University of Texas Press, 1978.
Clanchy, M. T. From Memory to Written Record. 1993. 2 ed.
Miller, William Ian. Bloodtaking and Peacemaking: Feud, Law, and Society in Saga Iceland. Chicago: University of Chicago Press, 1990.
Strayer, J.R. On the Medieval Origins of the Modern State. Princeton, 1970
In the medieval Islamic context, no. It was not true. The military caste of "lords" actually had very little authority to legislate at all. Whatever vagaries, complexities and disputes of interpretation may have existed among Islamic legal scholars, it still required at least some basis in the primary sources of jurisprudence-- the Quran, the Sunnah/Hadith, scholarly consensus, and lastly analogical reasoning.
I wrote about this in relation to the authorities of the caliph, specifically, in a recent post here: https://www.reddit.com/r/AskHistorians/comments/6tohxz/are_there_any_examples_of_the_ulama_rejecting_a/dlmmndh/?context=3
While the top answer and the other replies are great at emphasizing larger historiographic themes and particular areas of medieval law there doesn't seem to be an answer responding to the quote in the context of the question GRRM was answering on his blog.
GRRM's answer really concerns itself with inheritance law as he is really addressing a question about the murky waters of inheritance law in Westeros. As an example he grabs at the legal claims to the French Throne made in the Hundred Years War by Edward III and his successors. Edward III, through his mother Isabella of France, claimed the French throne after the death of Charles IV and the failure of the the senior male line of the Capetian dynasty.
He also uses another example of the succession crisis that rose from the history of Scotland after the death of King Alexander III. Without a clear male heir to the Scottish throne the crown was offered by the nobles of Scotland to Margaret of Norway whom died en route to Scotland. Leaving the succession even less clear than before. Resulting in disputes among the nobles of Scotland especially in the face of English and French interest seating one of their political allies. In such cases where there is no clear politically acceptable heir it seems that Martin is suggesting that political convenience or force of arms became the decider of whose claim to any particular throne was stronger and more legitimate.
I am not an expert in medieval inheritance or the crises that Martin uses to explain the realism of the way he depicted an murky inheritance in Westeros. But the quote clearly cannot be applied, no was it intended to, characterize all types of laws or sets of laws. Instead it seems fairly clear that he is suggesting that the vagueness of inheritance law might be a boon to powerful nobles to exploit in their favor. I would be hesitant to take the answer farther than he meant it as it would as other answers have pointed out characterize the ways in which legal systems formed and developed in the medieval world. Additionally, if anyone has a good answer specifically concerning inheritance I would like to see it.
If you'll forgive me I would like to address a broader cultural issue we seem to face about how we understand medieval life that is relevant to the idea of medieval lawlessness that Martin's quote seems to play into. I think there are numerous examples from medieval history that show us that there was a real desire and yearning for a concrete and enforceable legal process that would help create more stable and peaceful society.
A good example of this process would be the phenomena of the Peace and Truce of God in the tenth century and the broader conciliar movement that followed in the late medieval period.
Following the collapse of Carolingian power in the tenth century task of renegotiating concepts of governance, justice, and authority fell to new regional and local leaders from the Church and the Nobility whom had the power to exert control over their communities. The rapid decentralization of authority provided for an incredible upheaval that saw the rise of castellan lords who often challenged each other and the old nobility over regional control. This new era of violence and brooding regional conflict provoked a moral response from the Church and from within the nobility who had to grapple with their political incentives to participating violent conflicts with their religious obligations to not kill or murder.
The Peace and Truce of God attempted to regulate and set restrictions on when and how feuding could legitimately take place without risk of immortal damnation. Such efforts to reign in noble violence and participate in noble self-regulation are clear examples of the desire to create order out of morally problematic and chaotic situations that undermined the stability and safety of many. Far from being vague and contradictory, the Peace and Truce of God explained in detail what events and days combat was acceptable and who was off limits when violence broke out between neighboring lords.
Such self-regulation and the threat of heavenly repercussions to violating the regulations set down in the Peace and Truce are a kind of legal framework that sought to define and protect the vulnerable from physical violence and as a way to regulate the manner in which nobles would resolve disputes through violence.
The Peace and Truce of God, one example of many, shows that the medieval era was not a lawless period of violence and anarchy where nobles only sought more power and rarely submitted themselves to legal and moral authorities. The reality was quite the opposite. The existential desire to make sense of reality and to order their society in a way that reflected their moral values and remained flexible to the ever changing and evolving political, cultural, and economic realities is something we share with medieval historical actors.
It is true that many laws and legal codes as we know them did not exist in the medieval period but medieval societies did have laws, regulations, and complex ideas about acceptable behavior that are entirely contextual.
Sources:
Constance Bouchard, Strong of Body, Brave & Noble: Chivalry and Society in Medieval France
Richard Landes, The Peace of God: Social Violence and Religious Response in France Around the Year 1000
For a quick overview of the two examples cited by GRRM from the English perspective I would suggest:
Dan Jones, The Plantagenets: The Warrior Kings and Queens who Made England
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So kind of a follow up question here. What impact did law codices like the Corpus Juris Civilis have on the actual practice of laws in the countries that adopted them I suppose? In what ways did the legal systems change or adapt, and would this be an example that contradicts GRRM statement here? Would appreciate any thoughts.