Can $G$ (Newton's constant) be thought of as a fundamental parameter of physics? Can I think of Newton's Gravitational Constant as a fundamental parameter of nature (based on our current understanding of physics)?
I realize that it is very unlikely that $G$ is actually a fundamental parameter since, among other things, it has units. I am trying to come up with an easy-to-understand example of "a parameter of a model of nature" for laymen.
Can the speed of light be thought of as another "fundamental parameter of physics" for the purpose of this?
Are there any other examples of parameters of a physics model that would be easy to grasp for a non-science audience?
 A: 
Can I think of Newton's Gravitational Constant as a fundamental parameter of nature (based on our current understanding of physics)?


Can the speed of light be thought of as another "fundamental parameter of physics" for the purpose of this?

No, as surprising as it may seem, neither $c$ nor $G$ are considered fundamental constants simply because they have units. As such, their values don’t tell us about physics but about our units. In fact, this is how the SI units are defined in practice now.

I am trying to come up with an easy-to-understand example of "a parameter of a model of nature" for laymen.


Are there any other examples of parameters of a physics model that would be easy to grasp for a non-science majority?

If you want constants whose values tell us about physics rather than units then you need to look at unitless constants. Furthermore, you must exclude constants like $\pi$ whose value can be determined by pure mathematical computation. We need unitless constants whose value can only be determined experimentally. The most “layman friendly” of these is the fine structure constant $\alpha$.
$\alpha$ basically represents the strength of the electromagnetic force in a dimensionless way. $\alpha$ can be written in terms of constants with units as $$\alpha = \frac{k e^2}{c \hbar}$$
Very often when discussing the physical meaning of the constants with units people will talk about what would happen if one of them were doubled. But because of the relationship above it is impossible to change just one of these quantities. So, you must always specify at least two that change. Let’s therefore consider a very simplified example showing the importance of the dimensionless quantity:
In our example we will simply measure the length of a bar of metal in SI meters. The SI meter is based on the Caesium hyperfine transition and $c$. In terms of the quantities above the length of a meter is proportional to $\hbar/c\alpha^4$. Similarly, the length of a bar of metal scales with the Bohr radius so in terms of the quantities above it is proportional to $\hbar/c\alpha$.
So, first suppose that we double $c$ and halve $\hbar$. This will keep $\alpha$ fixed. The length of the meter will go down by a factor of 4, but so will the length of the metal bar. So the length of the metal bar measured in meters will be the same! Doubling $c$ and halving $\hbar$ will have no physical consequences because $\alpha$ was unchanged.
Now, suppose that we again double $c$ but halve $\alpha$. The length of the meter will increase by a factor of 8, but the length of the meter bar will stay the same. So the length of the metal bar measured in meters will go down by a factor of 8. Thus, a change in the unitless $\alpha$ is physically meaningful, while a change in the constants with units were not.
A: John Baez's article How Many Fundamental Constants Are There? is a pretty good explication of how physicists think about fundamental constants.  It's not quite written at the layman's level, though — more of the "advanced undergraduate" level.

You might at first think that the speed of light, Planck's constant and Newton's gravitational constant are great examples of fundamental physical constants.
But in fundamental physics, these constants are so important that lots of people use units where they all equal 1! The point is that we can choose units of length, time and mass however we want. That's three independent choices, so with a little luck we can use them to get our favorite three constants to equal 1.
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On the other hand, certain constants don't depend on the units we use - these are called "dimensionless" constants. Some of them are numbers like pi, e, and the golden ratio - purely mathematical constants, which anyone with a computer can calculate to as many decimal places as they want. But others - at present - can only be determined by experiment. These tell us facts about nature that are completely independent of our choices of units.
The most famous example is the "fine structure constant", $e^2/\hbar c$. Here $e$ is the electron charge, $\hbar$ is Planck's constant, and $c$ is the speed of light. If you work out the units involved you'll see it's dimensionless, and experiments show that it's about 1/137.03599. Nobody knows why it equals this. At present, it's a completely mysterious raw fact about the universe!
Constants that aren't dimensionless can be regarded as relating one sort of unit to another. For example, the speed of light has units of length over time, so it can be used to turn units of time (like years) into units of length (like light-years), or vice versa. People who are interested in fundamental physical constants usually start by doing this as much as possible - leaving the dimensionless constants, which are the really interesting ones.

A: Yes, $G$ is a fundamental constant in physics. The speed of light is also a fundamental constant, and so is the Planck constant.
A: There are just a few fundamental constants in Nature.. Like a Planck-length, or a Planck-second, a Planck-mass. Then we also have the coupling strength of the four (three, if gravity, which has mass as its source, which I already mentioned, is excluded) forces. More fundamental constants are not to be found.
Of course, $G$ (which also enters General Relativity) is constant, like my body temperature is too (under normal circumstances). And also is the speed of light,  $c$. Or Planck's constant, $h$. Or the spring constant $k$. Or whatever constant. But they can all be formed from the truly basic constants (units) I mentioned. These units (as the name already suggests) never vary (so they are constant). The non-fundamental constants can vary (in value) because our metric system has quite arbitrary units, like the meter, second, Joule, kilogram, etc. How can they be constants if they can vary?
If we express the non-fundamental constants in the fundamental constants I named, then these constants are the same for every intelligence in the Universe.
This means that all constants, setting the fundamental constants equal to one. Isn't that simple?
Except, of course, for the coupling strengths of the four forces. Obviously they can not all be equal to one.
So, tell laymen that $G$ is not really a fundamental constant. It's a constant, but not made up of fundamentals. And that its true value is one.
