  We consider a smooth, compact and embedded hypersurface $\Sigma$ without
boundary and show that the corresponding (shifted) surface Stokes operator
$\omega+A_{S,\Sigma}$ admits a bounded $H^\infty$-calculus with angle smaller
than $\pi/2$, provided $\omega>0$. As an application, we consider critical
spaces for the Navier-Stokes equations on the surface $\Sigma$. In case
$\Sigma$ is two-dimensional, we show that any solution with a divergence-free
initial value in $L_2(\Sigma,\mathsf{T}\Sigma)$ exists globally and converges
exponentially fast to an equilibrium, that is, to a Killing field.
