Why is water not part of the equilibrium constant? In the acid base reaction 
$$\ce{NH4+ + H2O &lt;-&gt; NH3 + H3O+}$$

the acidity constant, which is a relation between concentrations is given by 
$$K_{\mathrm{a}}=\frac {\ce{[NH3]} \cdot \ce{[H3O+]}}{\ce{[NH4+]} \cdot \ce{[H2O]}}$$

Since $\ce{[H2O]}$ is constant, it can be included inside the constant to obtain a new $K_\mathrm{a}$ which is $6.3 \times 10^{-10}$.

(Being a constant allows us to relate the molar concentration to the density of water.)

And so, even though some quantity of water reacts with $\ce{NH4+}$, but the volume of water varies in a way that $\ce{[H2O]}$ would still be constant.

What I don't understand is why the molar concentration of water is constant. Why isn't it handled the same way as the other reactant?
I understand that if $\ce{H2O}$ was the only liquid and the rest of the reactants and products were of different state (gaseous or solid), then water would exist alone in the beaker and its concentration would therefore remain constant, but this is not the case. 

That was about heterogeneousness. Trying another guess; in the reaction of esterification $$\ce{{alcohol} + {carboxylic acid} &lt;-&gt; {ester} + H2O}$$
$\ce{H2O}$ is considered as a product which concentration is not constant, so it is counted explicitly in the equilibrium constant.

Could the answer be that $\ce{NH4+}$, $\ce{NH3}$ and $\ce{H3O+}$ are dissolved in water, while the alcohol, the carboxylic acid, and the ester are not?

I'm looking for an answer that clearly shows which volume is considered (water or the solution), because I'm confused as to why something would or wouldn't be included in the equilibrium constant.
There is no IUPAC definition of $K_a$, however, many reputable texts like Levine's Physical Chemistry define $K_a$ as:

$$\frac {a(\ce{A-})a(\ce{H3O+})}{ a(\ce{HA})a(\ce{H2O})}$$

where "a(X)" is the activity of species "X".

Then various approximations can be made, such as approximating $a(\ce{H2O}) = 1$ and approximating the activity of the solutes as the concentration of the solutes.

These approximations are only reasonable in dilute solutions.