In Newtonian physics we could deduce conservation of energy, momentum and angular momentum from Newton's three laws.
But by Noether's theorem, conservation laws could be deduced from symmetries.
Could we deduce this conservation laws using only Galilean relativity without use of Newton's laws?
(Galilean relativity: all mechanical physical laws are same in all inertial frames)
No. Galilean invariance is about "how systems stay physically equivalent under galilean transformations" and Noether's theorem is about "what things stay the same along a dynamical trajectory". So you need to specify two things: what does it mean to be "physically equivalent" and what are the dynamics of the theory.
The approach is then to tackle the two at the same time: specify the dynamics by a second order ODE for the position and state that two systems are equivalent iff they lead to the same equations of motion (i.e if the dynamics is invariant under action of the galilean transformation). So you'll always need Newton's equations in this sense.
If you then restrict the problem to systems described by a stationary action principle, the equivalence relation becomes " two systems are physically equivalent iff $L^*-L=g'(t)$ for some $g$, where $L^*$ is the lagrangian of the second system". In these conditions you can prove several things (see, e.g, Landau and Lifshitz's Mechanics, vol.1), for instance, that the most general Galileo invariant Lagrangian for a free particle is $$L(x,v,t)=\frac{1}{2}m v^2$$
Now applying Noether's theorem you can deduce the usual conservation laws.
So, in short, you can deduce conservation laws from symmetries only if you specify the dynamics of the system (be it by a lagrangian, a hamiltonian, or Newton's laws).
