Can classical mechanics be axiomatized in a way similar to what is done in Euclidean geometry? Can classical mechanics be axiomatized in a way similar to what is done in Euclidean geometry?
If we took Newton's laws as axioms, what other axioms would be needed to deduce all of classical mechanics?
I read in https://sites.uwm.edu/nosonovs/2017/03/06/why-mechanics-is-a-fundamental-science-are-newtons-laws-laws-of-nature/ ("Why mechanics is a fundamental science: are Newton’s laws of nature?", by Michael Nosonovsky) that:

"Many people believe that all mechanics can be deduced logically from Newton’s laws in a manner similar to how geometry can be deduced from Euclid’s axioms. This is incorrect. By no means Newton’s laws constitute a self-sufficient set of logical axioms." 

I read in https://physicstoday.scitation.org/doi/10.1063/1.1825251 ("Whence the Force of F = ma? I: Culture Shock", by Frank Wilczek) that a "zeroth law of motion" would be necessary:

"[...] that mass is conserved. The mass of a body is supposed to be independent of its velocity and of any forces imposed on it; also total mass is neither created nor destroyed, but only redistributed, when bodies interact."

Can classical mechanics be presented axiomatically in a way similar to what is done in Euclidean geometry? Has it been done? What is the minimum number of axioms that would be needed? What could be taken as primitive notions? Etc...
 A: The point of the first paper is that $F = ma$ doesn't actually indicate how matter moves, at least not without specifying what $F$ is, which is correct. You can't really get much of use out of Newtonian mechanics, outside of general theorems, if you don't specify the forces. I don't think this is quite the takedown it is, because an axiom system can get by like that, but in that case the forces will have to go in the hypothesis. The issue of not specifying conservation of mass and so forth is also not very problematic, because that is also generally specified in more rigorous textbooks, where the system is defined by a set of $n$ particles as
\begin{equation}
\mathscr{P} = \{ (\vec{x}_i(t), m_i, e_i) \}_{1 \leq i \leq n}
\end{equation}
where you can input the trajectory, mass, charge, etc of each point particle. Given appropriate conditions on $x$, this implies the conservation of mass, particles, etc. The Newton law will then be something like 
\begin{equation}
\vec{F}_{i}(t) = m_i \ddot{\vec{x}}_i(t)
\end{equation}
You basically have two possibilities here. Either you define a specific classical theory, like point masses with gravity, in which case you can just use the laws, and your various initial conditions will be specific choices of $\mathscr{P}$, or you can take the general laws, but in that case if you want to actually compute something, you will have to define the forces involved. For instance, if you want to compute gravitational problems, you'll have to use the hypothesis

For any particle $i$, the force applied to it is $$F_{i} = \sum_{i \neq j} \frac{m_im_j}{\| \vec{x}_i(t) - \vec{x_j}(t) \|}$$
  and we have the initial conditions $x_i(0) = x_{i,0}$, $\dot{x}_i(0) = v_{i,0}$, and $m_i$ defined for every $i$.

It's a bit more work, but it doesn't fundamentally disqualify Newtonian laws from forming (along with a variety of other mathematical axioms) an axiom system. You can simply get all theorems by using every combination of possible forces, initial conditions and masses, although you might want to have a few conditions on the masses and forces (It's best to keep non-zero masses and forces that are regular enough).
You can define also classical mechanics in a way similar to general relativity using the Newton-Cartan theory, which is fairly well defined axiomatically.
If we go by the 100% axiomatized without any reference to other axiom systems, this has been done for Newton's law with gravity by H. Field in "Science without numbers", in a way similar to modern formulations of Euclid's axioms, ie the Hilbert axioms of geometry.
It is a fairly long text, and it doesn't have a recapitulation of all the axioms, but the rough sketch of system is thus : 
The object of the system are events $p, q, r, \ldots$ (similarly to the events in special relativity, these are equivalent to $p = (t, x, y, z)$, but coordinates aren't involved in the system), point particles, and scalar quantities. Point particles are defined by trajectories $S, T, \ldots$, which are roughly sets of every events that point particle go through. Scalar quantities can be a variety of things, in this case it's the gravitational potential.
The axioms are roughly the following :


*

*Some basic logical axioms, which are propositional logic and predicate logic with equality.

*Axioms regarding time. This involves three notions : that two events may or may not be simultaneous (ie $t = t'$), if they are not simultaneous, there is an ordering (either $t > t'$ or $t' > t$), and four events $(a,b,c,d)$ can be such that the time between $a$ and $b$ is the same as between $c$ and $d$ (time-congruence).

*The Hilbert axioms. If two events are simultaneous, every Hilbert axiom holds for these points.

*Axioms regarding scalar quantities, to define notions of between-ness ($\phi(y)$ is in between $\phi(x)$ and $\phi(z)$ if $\phi(x) < \phi(y) < \phi(z)$) and congruence (the notion of the potential being identical at two points).

*A very long list of axioms to define various operations one can do in a vector space. Just a lot of Hilbert axiom style definitions of products, ratios, derivatives, inner products and so forth.

*The laws of motion by considering the separation of two trajectories under the action of a gravitational field


The total number of axioms involved is quite long, depending on how you count them (a lot of them are just definitions). Depending on your axiomatization, there are about 4 axioms of propositional logic, 7 axioms for predicate logic with equality, 20 axioms for Hilbert, maybe 10 axioms involved in time, and the law of motion. Everything else can be roughly swept up under definitions, I think, although you may want to keep around everything related to the gravitational field. The book sort of putters off towards the end, and I think it's missing a few axioms (There's no axioms involving trajectories, such that no two events in a trajectory should be simultaneous, or that for every event, there should be one event simultaneous to it in the trajectory), so let's say a minimum of about 50 axioms. 
To give you an idea of the kind of formalism this is, here's the definition of the derivative. The directional derivative of a scalar field $\phi$ at $p$ in the direction $x_2 - x_1$ having the value $\phi(y_2) - \phi(y_1)$ is denoted by $D(p, x_1, x_2, y_1, y_2)$, and is defined by

Either $y_2 \approx y_1$ and $\exists y \{ \exists c \exists d$ ($y$
  is strictly Scal-between $c$ and $d$) and if $x_1 = x_2$ then $y
> \approx y$, and if $x_1 \neq x_2$, then $\forall c \forall d $(if $y$
  is strictly Scal-between $c$ and $d$ then there are points $a$ and $b$
  such that $ab \mathrm{Par} x_1x_2$ and $p$ is strictly st-between $a$
  and $b$ for all points $t$ other than $p$ that are strictly st-between
  $a$ and $b$, $(x_1x_2pt) \text{E-Bet}_{\mathrm{st, Scal}} (ptyc)
> (ptyd)$) $\}$ or $y_1 \not\approx y_2$ and $\exists x_3 \exists y_3$
  ($x_3 \text{st-Bet} x_1 x_2$ and $x_1 x_3 \text{P-Cong} x_3 x_2$ and
  $y_1 y_3 \text{Scal-cong} y_3 y_2$ and $\ldots$  )

I had to stop because it gets fairly long. You can 100% axiomatize classical mechanics like Euclidian geometry, but I wouldn't recommend it.
A: The axiomatic system you are looking for needs the mathematical axioms and theorems used for establishing differential equations and calculus,  to be complete, Newton's law's are not enough.
Euclidean geometry is simple, as it needs only algebraic relations as a mathematical framework. Classical mechanics needs a more sophisticated mathematical framework, which is complete, with its own axioms and theorems, with a huge number of possible solutions.
Newton's laws pick from the infinity of possible mathematical solutions those solutions that fit observational data, to start with, and, very important, are predictive of new situations.
Newton's laws are not enough, and it is very interesting that he had to invent calculus in order to model observations and measurements.
P.S. to understand the axioms of mathematics have a look at this answer here.
