  The renormalization of the vacuum energy in quantum field theory (QFT) is
usually plagued with theoretical conundrums related not only with the
renormalization procedure itself, but also with the fact that the final result
leads usually to very large (finite) contributions incompatible with the
measured value of $\Lambda$ in cosmology. Herein, we compute the zero-point
energy (ZPE) for a nonminimally coupled (massive) scalar field in FLRW
spacetime using the off-shell adiabatic renormalization technique employed in
previous work. The general off-shell result yields a smooth function $\rho_{\rm
vac}(H)$ made out of powers of the Hubble rate and/or of its time derivatives
involving different (even) adiabatic orders $\sim H^N$ ($N=0,2,4,6,...)$, i.e.
it leads, remarkably enough, to the running vacuum model (RVM) structure. We
have verified the same result from the effective action formalism and used it
to find the $\beta$-function of the running quantum vacuum. No undesired
contributions $\sim m^4$ from particle masses appear and hence no fine-tuning
of the parameters is needed in $\rho_{\rm vac}(H)$. Furthermore, we find that
the higher power $\sim H^6$ could naturally drive RVM-inflation in the early
universe. Our calculation also elucidates in detail the equation of state of
the quantum vacuum: it proves to be not exactly $-1$ and is moderately
dynamical. The form of $\rho_{\rm vac}(H)$ at low energies is also
characteristic of the RVM and consists of an additive term (the so-called
`cosmological constant') together with a small dynamical component $\sim \nu
H^2$ ($|\nu|\ll1$). Finally, we predict a slow ($\sim\ln H$) running of
Newton's gravitational coupling $G(H)$. The physical outcome of our
semiclassical QFT calculation is revealing: today's cosmic vacuum and the
gravitational strength should be both mildly dynamical.
