  Both the core collapse of rotating massive stars, and the coalescence of
neutron star (NS) binaries, result in the formation of a hot, differentially
rotating NS remnant. The timescales over which differential rotation is removed
by internal angular-momentum transport processes (`viscosity') has key
implications for the remnant's long-term stability and the NS equation-of-state
(EOS). Guided by a non-rotating model of a cooling proto-NS, we estimate the
dominant sources of viscosity using an externally imposed angular velocity
profile $\Omega(r)$. Although the magnetorotational instability provides the
dominant source of effective viscosity at large radii, convection and/or the
Spruit-Tayler dynamo dominate in the core of merger remnants where $d\Omega/dr
\geq 0$. Furthermore, the viscous timescale in the remnant core is sufficiently
short that solid body rotation will be enforced faster than matter is accreted
from rotationally-supported outer layers. Guided by these results, we develop a
toy model for how the merger remnant core grows in mass and angular momentum
due to accretion. We find that merger remnants with sufficiently massive and
slowly rotating initial cores may collapse to black holes via envelope
accretion, even when the total remnant mass is less than the usually considered
threshold $\approx 1.2 M_{\rm TOV}$ for forming a stable solid-body rotating NS
remnant (where $M_{\rm TOV}$ is the maximum non-rotating NS mass supported by
the EOS). This qualitatively new picture of the post-merger remnant evolution
and stability criterion has important implications for the expected
electromagnetic counterparts from binary NS mergers and for multi-messenger
constraints on the NS EOS.
