  We present a class of generally covariant nonlocal gravity models which have
a flat-space general relativistic (GR) limit and also possess a stable de
Sitter (dS) or Anti-de Sitter (AdS) background with an arbitrary value of its
cosmological constant. The nonlocal action of the theory is formulated in the
Euclidean signature spacetime and is understood as an approximation to the
quantum effective action (generating functional of one-particle irreducible
diagrams) originating from fundamental quantum gravity theory. Using the known
relation between the Schwinger-Keldysh technique for quantum expectation values
and the Euclidean quantum field theory we derive from this action the {\em
causal} effective equations of motion for mean value of the metric field in the
physical Lorentzian-signature spacetime. Thus we show that the (A)dS background
of the theory carries as free propagating modes massless gravitons having two
polarizations identical to those of the Einstein theory with a cosmological
term. The on-shell action of the theory is vanishing both for the flat-space
and (A)dS backgrounds which play the role of stable vacua underlying
respectively the ultraviolet and infrared phases of the theory. We also obtain
linearized gravitational potentials of compact matter sources and show that in
the infrared (A)dS phase their effective gravitational coupling $G_{\rm eff}$
can be essentially different from the Newton gravitational constant $G_N$ of
the short-distance GR phase. When $G_{\rm eff}\gg G_N$ the (A)dS phase can be
regarded as a strongly coupled infrared modification of Einstein theory not
only describing the dark energy mechanism of cosmic acceleration but also
simulating the dark matter phenomenon by enhanced gravitational attraction at
long distances.
