  We compute by supersymmetric localization the expectation values of half-BPS
't~Hooft line operators in $\mathcal{N}=2$ $U(N)$, $SO(N)$ and $USp(N)$ gauge
theories on $S^1 \times \mathbb{R}^3$ with an $\Omega$-deformation. We evaluate
the non-perturbative contributions due to monopole screening by calculating the
supersymmetric indices of the corresponding supersymmetric quantum mechanics,
which we obtain by realizing the gauge theories and the 't~Hooft operators
using branes and orientifolds in type II string theories.
