  We study the nonlinear gravitational dynamics of a universe filled with a
pressureless fluid and a cosmological constant $\Lambda$ in the context of
Newtonian gravity, and in the relativistic post-Friedmann approach proposed in
paper I [I. Milillo et al., Phys. Rev. D 92, 023519 (2015).]. The
post-Friedmann approximation scheme is based on the $1/c$ expansion of the
space-time metric and the energy-momentum tensor, and includes nonlinear
Newtonian cosmology. Here we establish the nonlinear post-Friedmann framework
in the Lagrangian-coordinates approach for structure formation. For this we
first identify a Lagrangian gauge which is suitable for incorporating nonzero
vorticity. We analyze our results in two limits: at the leading order we
recover the fully nonlinear Newtonian cosmological equations in the Lagrangian
formulation, and we provide a space-time metric consistent from the perspective
of general relativity. We then linearize our expressions and recover the
relativistic results at first order in cosmological perturbation theory.
Therefore, the introduced approximation scheme provides a unified treatment for
the two leading-order regimes, from the small scales described by Newtonian
gravity to the large linear scale, where first-order relativistic cosmological
perturbation theory gives a very good description of structure formation.
