  The gas needed to sustain star formation in galaxies is supplied by the
circumgalactic medium (CGM), which in turn is affected by accretion from large
scales. In a series of two papers, we examine the interplay between a galaxy's
ambient CGM and central star formation within the context of the large-scale
environment. We use the IllustrisTNG-100 simulation to show that the influence
exerted by the large-scale galaxy environment on the CGM gas angular momentum
results in either enhanced (Paper I) or suppressed (Paper II, this paper) star
formation inside a galaxy. We find that for present-day quenched galaxies, both
the large-scale environments and the ambient CGM have always had higher angular
momenta throughout their evolutionary history since at least $z=2$, in
comparison to those around present-day star-forming disk galaxies, resulting in
less efficient gas inflow into the central star-forming gas reservoirs. A
sufficiently high CGM angular momentum, as inherited from the larger-scale
environment, is thus an important factor in keeping a galaxy quenched, once it
is quenched. The process above naturally renders two key observational
signatures: (1) a coherent rotation pattern existing across multiple distances
from the large-scale galaxy environment, to the circumgalactic gas, to the
central stellar disk; and (2) an anti-correlation between galaxy star-formation
rates and orbital angular momenta of interacting galaxy pairs or groups.
