What would happen if $F=m\dot{a}$? What would happen if instead of $F=m \frac{d^2x}{dt^2}$, we had $F=m \frac{d^3x}{dt^3}$ or higher?
Intuitively, I have always seen a justification for $\sim 1/r^2$ forces as the "forces being divided equally over the area of a sphere of radius $r$".
But why $n=2$ in $F=m\frac{d^nx}{dt^n}$ ?
 A: Because the second derivative is fully determined by the external force, it is an experimental fact. The force can be directly measured with a dynamo-meter. It is not an abstraction. So the acceleration is known as soon as the force is known and vice versa. The trajectory depends also on the initial conditions which are independent from the force but reference frame-dependent.
In CED there is an equation (Lorentz-Abraham one) with a third derivative in time. It has non physical (runaway) solutions.
A: Rephrasing in a Lagrangian manner, this is equivalent to asking about higher-derivative theories. See: Why are there only derivatives to the first order in the Lagrangian?
A: Another reason is that if n did not equal 2, some of the symmetries of the Newton equation would be lost. For example, classically, physics on a microscopic scale is time reversal invariant. We can see this from the Newton equation because if x(t) is sent to x(-t), the 2nd time derivative ensures that the negatives cancel. If n was an odd number, we would not observe this symmetry.
If n was smaller than 2, then Galilean transformations would not leave the equation invariant. If we send x(t) to x(t) + vt, then the second time derivative kills the vt term on the LHS. If n=1, then this would not be possible, and relative velocity would be meaningless (even non-relativistically). If n was some number larger than 2, then transformations that send x(t) to x(t) + b(t^m), where m is less than n, would be a symmetry of the Newton equation. But relative accelerations, jerks, etc. produce observable discrepancies, so that should not be possible either.
Again, this may not really answer "why" in the sense that you are asking, but the fact that the equation matches observations is enough in science to justify it. The Newton equation is basically experimental fact, like Vladimir says. 
A: It is because the evolution of mechanical system is fully determined by initial coordinates and speeds. Therefore, your equation must be of second order, otherwise setting initial accelerations and "speeds of increase of accelerations" would be necessary.  
A: Newton second law is known as fundamental law of mechanics, because it is supposed to solve the fundamental problem of mechanics, that is, finding the position of a particle at any given moment in time, i.e., to find 
$$
x=f(t)
$$
Plot of $x=f(t)$ can only be a straight line (a special type of curve with curvature=0) or a curve (any curve with curvature <>0). However, any plot just depends on a starting point (current value or initial value) and on how it changes when moving forward or backwards from from that given point. Such a change is represented by curve’s curvature, i.e, from this point on, your choices are, continue straight ahead (curvature=0), go up (curvature>0) or go down (curvature<0). How much you go down or up, dependes on curvature’s magnitude.
It happens that curvature depends only on second and first order derivatives 
$$
\textrm{curvature}=\frac{x^{\prime\prime}}{\left(1+{x^\prime}^2\right)^{3/2}}
$$
So, any possible curve for $x=f(t)$ would just be characterized by its first and second derivatives provided that force in $F(x’,x,t)=m \frac{d^2x}{dt^2}$is properly defined.
In an universe with higher order derivative (with respect to us), one could always set that universe’s straight line to be the solution of our $n^{th}-1$ derivative, meaning that in that particular universe Newton’s first law would be a curve with respect to us, but not with respect to themselves, and all they would need to discriminate their natural state of motion (motion in a systems straight line) would be that universe’s second order derivative.
In summary second order derivative is all one needs to differentiate natural states of motion with affected states of motion.
One needs to understand that even though many quantities like, energy, momentum, velocity, acceleration, force, jerk and so on… are (and may be) defined in mechanics being afterwords useful in other branches of science, the ultimate goal of mechanics is to find $x=f(t)$.
A: In statics, you can still have a force without acceleration so $F$ is independent of $a$. $F$ is the cause of the change in the position of an object initially at rest in some frame. To give it physical meaning, you have to define how it's to be measured and one way would be to define 1 unit of F causing one unit of compression in some standard spring.
Now if $F$ causes a body at rest to change its position, then over a time dt the postion has changed by dx. Your job as a physicist is to construct an equation relating F to the change in velocity of the body.
So with all this in mind, what would happen if $F=m*d^3x/dt^3$ ?
It would mean that even though $F$ is the cause behind the change in velocity of a body, there are some changes in the velocity possible where $F = 0$ such as for $a = const$. You would end up with particles accelerating in arbitary directions for $F = 0$.
A: There is a deeper reason for $F~=~\frac{d^2x}{dt^2}$  Within the Galilean group it is an invariant with respect to all changes of frame $x’~=~x~+~vt$.  The acceleration of a body is not something which can be made to vanish by boosting to another Galilean frame as
$$
F’~=~\frac{d^2x’}{dt^2}~=~\frac{d^2x}{dt^2}~+~\frac{d^2vt}{dt^2}
$$
where for constant $v$ the second term is clearly zero.  The next higher derivative $dF/dt~=~mda/dt$, called a jerk” is also invariant, as are all $d^nx/dt^n$, but the acceleration contained in $T^2_p$ is the lowest element on the jet $T^n_p$ $n~\ge~2$ which is invariant.  Further, odd powers of $n$ would not be time reverse invariant under $t~\rightarrow~-t$
