What is an example of a quantitative analysis for the lift using Newtonian mechanics?
The governing equations of fluid mechanics, Navier-Stokes equations, are derived based on Newtonian mechanics (plus thermodynamics). The circulation theory of lift is derived using a simplified form of N-S equations. It's powerful because it's predictive for important things like lift and induced drag. The Newtonian explanation is not that useful because it's not predictive. Good for party talks, but poor for engineering.

That being said, if you couple it with the concept of control volume and mass conservation, you can derive something. Take a look at NASA Memo 1-16-59L. To summarize, as air moves past the wing, it gets deflected, so the lift in the vertical direction (normal to the incoming flow direction) is:

$$L=\dot{m}\Delta V$$ 

where $\Delta V$ is the portion of air that was induced across the wing interface in the vertical direction, $\dot{m}$ is the total mass flow rate of air through the wing. That's the Newtonian part of the story.

Now from the conservation of mass, what comes in = what goes out. So we can express mass flow rate as, assuming the wing is like a giant circle with diameter equal to its span (note that this assumption is the major pitfall of this analysis, since there is no reason the stream-tube has to be the diameter of the wing):

$$\dot{m}=\rho A_iV_{i}=\rho\pi(\frac{b}{2})^2V_{\infty}$$

where $b$ is the wing span, $V_{\infty}$ is the freestream airspeed, $\rho$ is air density.

As for the change in vertical airspeed, we can parametrize it by assuming the whole stream tube gets deflected by the wing through an angle $\epsilon$. Therefore, $\Delta V=V_{\infty}\sin \epsilon$, and:

$$L=\frac{\pi}{4}b^2\rho V_{\infty}^2 \sin \epsilon$$

That's pretty close to the lift equation that we know. In fact, if you express drag as the loss of horizontal airspeed, apply small angle approximation to the second order, then normalize the lift and drag by $\frac{1}{2}\rho V^2 S_{ref}$, you get (with $S_{ref}$ being the reference wing area, $A$ being the aspect ratio):

$$C_{D}=\frac{C_L^2}{\pi A}$$

That's the induced drag expression for an elliptical lift distribution that we would get from the circulation theory of lift. But that's pretty much as far as you can go. You have no way of estimating $\epsilon$ from first principle, so the theory isn't all that predictive.