No, the three laws are independent.
- The 1st law does not follow from the 2nd
The point here is that you have to understand what Newton meant by "force". For Newton, fictitious forces, i.e. forces which arise in accelerating reference frames, are not forces.
For example, if you are in an accelerating car, you will experience an acceleration in the direction opposite to that of the car's. But this acceleration is not caused by a (Newtonian) force.
So, for Newton, force implies acceleration, but acceleration does not imply force.
In modern terms, the first two laws would be formulated in the following way:
First law: When viewed in an inertial reference frame, an object either remains at rest or continues to move at a constant velocity, unless acted upon by a net force.
Second law: In an inertial reference frame, the vector sum of the forces F on an object is equal to the time derivative of its momentum: $\vec F= \dot{\vec{p}}$.
Notice the first words in the statement of the second law: in an inertial reference frame. But what is an inertial reference frame? It is that which is defined by the first law. So, in modern terms, we would say that the first law defines the inertial reference frames, while the second law tells us how motion (momentum) and force are related in such frames.
Could we include fictitious forces in the second law to get rid of the first? Maybe. But this is not how Newton formulated the laws, and could result in a lot of complications.
For a nice discussion, see this article.
- The 3rd law does not follow from the second, either
If we consider a system of two point masses on which no external force is acting, we have, from the 2nd law (let's drop the vector notation for simplicity):
$$F=\frac{d}{dt} (p_1+p_2) =0 \to \frac{dp_1}{dt}+\frac{dp_2}{dt} = 0\\ \to F_1 = - F_2$$
Let's indicate with the notation $F_{ij}$ the force caused on particle $i$ by particle $j$. Since there is no external force, the force acting on particle 1 can come only from particle 2: $F_1=F_{12}$. The same is true for particle 2, so that we obtain
$$F_{12}=-F_{21}$$
So we were able to derive the 3rd law from the 2nd!
...Weren't we?
No. Consider now three particles: we would get
$$F_1+F_2+F_3 =0 \to (F_{12}+F_{13})+(F_{21}+F_{23})+(F_{31}+F_{32})=0$$
That is to say
$$\sum_{ij} F_{ij}=0$$
Of course, $F_{ij}=-F_{ji}$ (Newton's third law) is a solution of this equation...but it is not the only one!
So, Newton's third law is not a consequence of the second.