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In this second paper we define a Post-Minkowskian (PM) weak fieldapproximation leading to a linearization of the Hamilton equations of ADMtetrad gravity in the York canonical basis in a family of non-harmonic3-orthogonal Schwinger time gauges. The York time ${}^3K$ (the relativisticinertial gauge variable, not existing in Newtonian gravity, parametrizing thefamily and connected to the freedom in clock synchronization, i.e. to thedefinition of the instantaneous 3-spaces) is put equal to an arbitrarynumerical function. A ultraviolet cutoff on the matter allows a consistentlinearization, which is shown to be the lowest order of a HamiltonianPost-Minkowskian (HPM) expansion. We solve the constraints and the Hamiltonequations for the tidal variables and we find Post-Minkowskian gravitationalwaves with asymptotic background (and the correct quadrupole emission formulaand energy balance) propagating on dynamically determined non-Euclidean3-spaces in these non-harmonic gauges. In conclusion we get a PM solution forthe gravitational field and we identify a class of PM Einstein space-times,which will be studied in more detail in a third paper together with the PMequations of motion for the particles and their Post-Newtonian expansion (butin absence of the electro-magnetic field). Finally we make a discussion on the{\it gauge problem in general relativity} to understand which type ofexperimental observations may lead to a preferred choice for the inertial gaugevariable ${}^3K$ in the PM space-times. In the third paper we will show thatthis choice is connected with the problem of dark matter.

The Einstein-Maxwell-Particle System: II) The Weak Field Approximation  in the Non-Harmonic 3-Orthogonal Gauges and Hamiltonian Post-Minkowskian  Gravity: the N-Body Problem and Gravitational Waves with Asymptotic  Background