In my opinion, the most fundamental law of Newtonian physics is the classical limit of the most fundamental law of quantum physics, namely the Feynman sum-over-all-histories principle (also known as the Feynman path integral).
Feynman discovered that the probability amplitude for a quantum system to evolve from state A to state B is proportional to $$\sum_{A \rightarrow B}e^{iS(A\rightarrow B)/\hbar}$$ where the "action" $S(A\rightarrow B)$ is a number that depends on how the system evolves between A and B and $\hbar$ is Planck's constant divided by $2\pi$. (The actual probability is proportional to the square of the magnitude of this sum, and the proportionality constant is determined by making the probabilities sum to 1.) With the proper choice of how to compute $S$, this explains quantum systems from the hydrogen atom to the Standard Model of particle physics.
In the limit $\hbar\rightarrow 0$, the sum-over-all-histories principle becomes the classical Principle of Stationary Action, $\delta S = 0$, which says that the classical path from A to B is the one from which deviations cause no first-order variation in the action. When the action is written as a time-integral of a Lagrangian, this principle is equivalent to the Euler-Lagrange equations.
So a single unifying principle lies behind quantum mechanics, quantum field theory (including the Standard Model), Newtonian mechanics, relativistic classical mechanics, electromagnetism, and General Relativity! I consider it the deepest of all known laws of physics.
This leaves the non-trivial question of how to compute $S$. We already know what $S$ is for Newtonian mechanics, for relativistic classical mechanics, for non-relativistic quantum mechanics, for electromagnetism, for General Relativity, and for the Standard Model.
The most important thing that stands out about the various expressions for $S$ that we've found is that they are invariant under various symmetries. The symmetries of $S$ are what give rise to the conservation laws. So we know how to invent new possible actions that might be the right one for, say, quantum gravity. The search for the "ultimate" action that describes a Theory of Everything is underway, under the assumption that it will be even more symmetric than the actions we already know about.
In short: The concept of "action" is what unifies physics. The search for the right action to unify all phenomena is what physics is about.
As a specific example, to learn about Newtonian gravity in this way, start with the action
$$S=\int_{t_1}^{t_2}(K-U)dt=\int_{t_1}^{t_2}\left[\frac{1}{2}m\left(\frac{d\vec{r}(t)}{dt}\right)^2+\frac{GMm}{|\vec{r}(t)|}\right]dt$$
for a small test mass $m$ moving under gravitational field of a much larger mass $M$ (assumed to be stationary at the origin). It happens to be the time integral of the kinetic energy of the system minus its potential energy. Show that $\delta S=0$ leads to elliptical orbits, and that energy and angular momentum are conserved.
