Given that a collection of classical particles can be modelled using Newton's three laws, it must be the case that both the conservation of momentum and the conservation of angular momentum are emergent features of Newton's three laws.
I can easily show the conservation of linear momentum from the third law, but how can the conservation of angular momentum be derived from Newton's three laws?
Deriving the rotational form of $\mathbf{\vec{F} = m \vec{a}}$ (as per request in comments)
Begin with Newton's second law of motion
$$\vec{F} = m\vec{a}.$$
Multiply both sides by a vector cross product with position gives
$$ \underbrace{\vec{r} \times \vec{F}}_{\text{Definition of torque}} = \vec{r} \times m \vec{a}. $$
Using the definition of the cross product, the above equation can be equivalently expressed as
$$ |\vec{r} \times \vec{F}| = rma \; \sin{\theta}. $$
Notice that since the force and acceleration are parallel we may consider $a \sin \theta$ as the tangential acceleration $a_t$. Finally this can be cast into final form given by
$$\tau = mr^2 \left(a_t/r \right),$$
where moment of inertia and angular acceleration are given by $$I=mr^2, \;\; \alpha = a_t/r,$$ respectively.
Answer:
You can consider the torque equation $$\vec{\tau} = I \vec{\alpha},$$ as the rotational equivalent to Newton's second law of motion where torque, moment of inertia and angular acceleration are given by $\vec{\tau}, \; I$ and $\vec{\alpha}$ respectively.
Now notice that angular acceleration is the time derivative of angular velocity $$ \vec{\tau} = \frac{d (I \vec{\omega})}{dt}.$$ which can be re-arranged into integral form $$\int\vec{\tau} dt= \int d(I \vec{\omega}).$$
Angular momentum $\vec{L}$ is given by the relation $\vec{L}=I\vec{\omega}$. Finally, in Newtonian mechanics, the conservation of angular momentum is used in analysing central force problems i.e. forces in the radial direction. With this is mind, I quote the definition of torque in terms of force
$$\vec{\tau} = \vec{r} \times \vec{F},$$ where the position and net force acting on a body are given by $\vec{r}$ and $\vec{F}$ respectively. If the force on some body is constantly pointed in the same direction then there is no torque on said body. Hence $$ C = I \omega = L, $$ and hence, $$ \frac{d L}{dt} \equiv 0. $$
We repeatedly use the fact that a cross product of parallel vectors vanishes. Suppose body $i$ has position $r_i$ and momentum $p_i\parallel\dot{r}_i$, and angular momentum $L_I=r_i\times p_i$ so$$\dot{L}_I=\underbrace{\dot{r}_i\times p_i}_{0}+r_i\times \dot{p}_i=r_i\times \dot{p}_i.$$If body $j$ exerts a force $F_{ij}$ on body $i$, $\sum_i\dot{L}_i=\sum_{ij}r_i\times F_{ij}$.By Newton's third law, this is $\frac{1}{2}\sum_{ij}\left(r_i-r_j\right) \times F_{ij}$. The $ij$ term vanishes if $F_{ij}$ is radial.
