These are of two sorts, which I will call top-down and bottom-up. I'll do the latter first.
Bottom-up argument
The fundamental and basic point of natural law jurisprudence is that law is the authoritative promulgation of dictates of practical reason (this isn't a complete definition of law, but the primary element to which natural law jurisprudence appeals). Every act of reasoning is based on principles. This is true of practical reasoning as well; our practical reasoning gets its force through conformity with certain principles that are fundamental and naturally known. For instance, if a line of reasoning forces us to the conclusion that something evil should be done, we know that we have gone wrong somewhere and that our reasoning violates a fundamental principle (evil should not be done). Such reasoning is not according to reason, in that it is not according to the fundamental principles to which reasoning must be conformed in order to be genuinely rational. Thus there are natural principles of reason to which practical reasoning must conform in order to be rationally authoritative. Natural law consists of those principles to which practical reasoning (particularly about things to which we are naturally inclined, like living in society) must conform in order to be according to reason.
It is natural law, therefore, that makes possible obedience to positive law in the first place, since it is natural law that makes possible all practical reasoning (which obedience to positive law presupposes). It is natural law, then, that justifies the creation and preservation of the legal mechanisms of civil society through which positive law is possible, since it contains the principles that govern practical reasoning about what would be conducive to the common good of each us insofar as we are interacting with each other in society. It is natural law that gives us the reason why law is binding in the first place. It is natural law that begins to establish a commonality of law among nations. This commonality need not be perfect; since positive law presupposes reasoning, and often a lot of reasoning, some nations may have reasoned more carefully or more extensively along a particular line than others. But the commonality does exist, and is in the ordinary work of society generally recognized to exist. This is what is called ius gentium, the collection of principles recognized by societies generally; the more civilized the society is, the more they are recognized.
It is also natural law that makes possible a real international law, since the first basics of a law of nations derive from the need for interactions between peoples or nations to be in conformity with the basic principles of practical reason. It is possible, therefore, to have a real law of nations independent of the particular legal mechanisms that are responsible for civil law, a basic framework that allows nations to interact practically for common good. This framework is also generally recognized to exist in the ordinary interactions of various peoples and nations. It is this framework that is already in place to justify the extension of legal mechanisms analogous to those in civil cases (treaties, for instance) to international cases, in a way that allows those mechanisms to be binding at all. International law, therefore, in both its customary and treaty-mechanism forms, is itself an evidence of natural law.
This allows an additional consideration. Through natural law we are able to hold governments accountable by determining whether they are acting in accordance with the principles and conclusions of practical reasoning that make them possible in the first place. An example of this is appeal to natural rights against the tyrannies of government. Natural rights are more fundamental than any positive law precisely because they are upstream (as it were) from positive in practical reasoning. In virtue of these elements of our practical reasoning, people can demand of the sovereign, demand of the legislative powers that constitute government, that they not act in certain ways; people can insist that civil rights should be distributed in certain ways. This is because natural rights are elements of something more fundamental than positive law itself, something that grounds the very possibility of government and civil society: natural law, constituted by the fundamental principles of practical, moral, and legal reasoning itself. Natural law is capable of being a bulwark against tyranny and oppression because it transcends, rationally and morally, all mechanisms of society.
Top-down argument
Granted that the world is ruled by divine providence, it is ordered according to promulgated dictates of divine reason; the whole community of the universe, as Aquinas says, is governed by divine reason. Thus there is an eternal law according to which the world is governed. Since all things are governed in some way by this law, everything in its own fashion participates in this eternal law. Rational creatures are also subject to the eternal law involved in divine providence; they participate in this law rationally. Since they participate in the rule of divine reason according to their own reason, they participate in providence itself through principles of reason that govern how they guide their own practice and the practices of others. This participation, these principles of reason, is what we call natural law.
