Can Natural Law Save A Non-Christian?

No. Natural love cannot save anyone, and all that the natural law dictates is natural love. Nor can the law give the power to relate to God as His adopted son, unless the law is the Power of God in the Heart; then the law is the power. The natural law has not the power; nor has the Mosaic law the power to save. Moses came with the law; but Jesus came with Grace and Truth, with power. That power is the Holy Spirit! (Jn 1; Rom 8; etc.).

Can someone outside the Covenant be saved? More precise question, calling for more precise answer. In response to a query, I have written the below answer, touching on the Covenant, Church, grace outside the sacraments, and why evangelize? MY RESPONSE:

I have to be succinct with the answers to your excellent questions. I hope it does not come across to bluntly. I could spend hours with you discussing and very happy to do so in person. But to cut to the chase:
It is a dogma that one is saved if and only if one dies in the state of grace. Thus, the law (Mosaic or natural law) cannot save me. It has not the power. This is basic Pauline teaching (Rom 3ff) which contextualizes what he means in Rom 2. So, Paul does not mean that one can without grace be saved by trying to adhere to the natural law. Neither did Pius IX when he echoed Paul. Rather, Pius IX means that outside the Church, one can receive the grace necessary for salvation. Further, someone visibly outside the Church can even cooperate with these graces and be saved. However, is this happening in great percentages? That is another matter, and answers should be sober and balanced.
It would be imbalanced to say it never happens and imbalanced to say it happens pretty much in most cases. What is the balanced statement?
The ordinary gateway to salvation is through the sacraments, the works of Christ the High Priest, sending gifts to men through his appointed ministers in the appointed ways. E.g., after committing a mortal sin, most Catholics recognize that the only way they can have confidence of re-entry into grace is through Confession. The Church teaches that to fail to take the opportunity to get to confession in such a state is implicit contempt for the sacrament and thus implicit (but real) rejection of grace.
It is also confirmed teaching that grace is available outside the sacraments. This, however, is ordered to the reception of grace through the sacraments. It may be the good thought Lydia had to continue to listen to Paul and to be baptized, her and her family (children too?). So, the grace makes a turning possible, and the turning leads to sacraments, and the sacraments are the actual entryway.
It is not impossible that someone who never hears of or reaches the sacraments can be saved. However, unspeakably more grace is available through the sacraments. And so, the likelihood of salvation outside the Church is very uncertain, unsteady, unreliable. Pius XII, Mystici corporis is excellent on this. Also, hidden grace is ordered to the sacraments, and those who enjoy the sacraments have a universal mission to spread the word. Those of good will who secretly receive grace will be disposed to accept the message rightly preached. Of course, one can never judge the interior heart. But one can judge character. The fact that many reject the message rightly preached is indication that they are not on the way to salvation.
There is current euphoria over the possibility that “all” may be saved. The opinion is wildly erroneous. The euphoria is pastorally destructive, not ennobling and inspiring. Revelation is trampled and dogmas ignored. With this as the ultimate dramatic backdrop, a kind of ultimate entitlement (Who is God to Judge Me?!) the West has fallen into indifference about Christ and his religion and about living an upright moral life. Meanwhile, certain moral sensitivities, while having genuine goods to protect, have gone berserk in their intensity and myopia (tobacco, pollution, migration) and made people lose even judgment and common sense, labeling all who would strive for balance and common sense to be haters. I have another blog post on the issue of this euphoria of universal salvation entitled “Balthasar’s Delirious Hope”; it goes into a little more detail concerning the theologian partly behind this euphoria.
There is yet another issue that touches on the theme of your questions.
One needs to distinguish the essence of a religion from its elements. The essence is the package deal, how it coheres as a whole and whether or not it is duly appointed. There is only one religion that is true and duly appointed today, and it is the Catholic religion. Other religions have some elements of truth and also some elements of error. However, no other religion today is appointed by God as the way of salvation. When we simply compare elements, we speak of “fulness of truth” in the Catholic religion and “degrees of approximation” to that in other Christian communities and even other religions. These are true assertions: fulness here and degrees outside. However, to compare the elements is not the only relevant analysis. Another, much more essential and significant, analysis regards the question of which religion is appointed by God. On this analysis, only one religion is true, the Catholic religion. Sadly, this analysis has not been actively employed much in the past five decades, but its truth has not disappeared. In fact, this truth is even one of the “elements”. See my post on the “Fulness of Truth” for more.
Why be Catholic? If one believes it is the true religion appointed by God, one should be Catholic. One can make an argument for this; I have sketched one in a podcast On the One True Religion.
The Catholic religion is not spread by the sword (although it may be protected by the sword). It is spread best by a loving presentation that does not refrain from the “whole truth”. Should I leave the Catholic Church, I would be aware of rejecting the grace of God. Should I fail to present it to another who is receptive, I believe I have wronged that person. I might fail, but God asks me.