  We construct the simplest solutions of the previously obtained precanonical
Schr\"odinger equation for quantum gravity, which correspond to the plane waves
on the spin connection bundle and reproduce the Minkowski spacetime on average.
Quantum fluctuations lead to the emergence of the minimal acceleration $a_0$
related to the range of the Yukawa modes in the fibers of the spin connection
bundle. This minimal acceleration is proportional to the square root of the
cosmological constant $\Lambda$ generated by the operator re-ordering in the
precanonical Schr\"odinger equation. Thus the mysterious connection between the
minimal acceleration $a_0$ in the dynamics of galaxies as described by
Milgrom's MOND and the cosmological constant emerges as an elementary effect of
precanonical quantum gravity. We also argue that the observable values of $a_0$
and $\Lambda$ can be obtained when the scale of the parameter $\varkappa$
introduced by precanonical quantization is subnuclear, in agreement with the
previously established connection between the scale of $\varkappa$ and the mass
gap in quantum SU(2) Yang-Mills theory.
