What is the connection between Newton's laws of motion and his law of gravitation? What is the connection between Newton's laws of motion and his law of gravitation? Is there a connection where one suggests the other, or are they just two separate ideas that are assumed to be true?  
 A: The $F= \frac{GMm}{r^2}$ law (Or equivalently $U=-\frac{G M m}{r}$ potential law) can't be deduced from Newton's laws of motion. It's a little secret that parts of physics like this are, from a purely mathematical perspective, totally arbitrary.
For example, it's perfectly reasonable to play build-a-reality and plug in whatever interaction potentials you want. $U=\frac{A}{r^{12}}-\frac{B}{r^6}$ is a completely valid interaction in classical mechanics. It doesn't contradict anything.
In more modern physics the rules can be much more restrictive. (which is a good thing, because it means you can't just plug whatever you want into the theory!)
A: Newton's Laws of Motion provide a framework for all types of forces, connecting observed motions to the forces that control them.
Newton's Universal Law of Gravitation tells you how to calculate the force of gravity. So if you want to solve a problem with gravity you use this law for the force, and the laws of motion to solve the problem.
Many practical problems include multiple forces, thus thre are force laws for friction, springs, etc. Some of these are just good approximations. Engineering uses all of them.
