We revisit the problem of matching the strong coupling expansion of the
$\frac{1}{2}$ BPS circular Wilson loops in ${\cal N}=4$ SYM and ABJM gauge
theories with their string theory duals in ${\rm AdS}_5 \times S^5$ and ${\rm
AdS}_4 \times CP^3$, at the first subleading (one-loop) order of the expansion
around the minimal surface. We observe that, including the overall factor
$1/g_{\rm s}$ of the inverse string coupling constant, as appropriate for the
open string partition function with disk topology, and a universal prefactor
proportional to the square root of the string tension $T$, both the SYM and
ABJM results precisely match the string theory prediction. We provide an
explanation of the origin of the $\sqrt T$ prefactor based on special features
of the combination of one-loop determinants appearing in the string partition
function. The latter also implies a natural generalization $Z_\chi \sim (\sqrt
T/g_{\rm s})^\chi$ to higher genus contributions with the Euler number $\chi$,
which is consistent with the structure of the $1/N$ corrections found on the
gauge theory side.