  A detailed five-dimensional calculation of the Higgs-boson decay into two
photons is performed in both the minimal and the custodially protected
Randall-Sundrum (RS) model, where the Standard Model (SM) fields propagate in
the bulk and the scalar sector lives on or near the IR brane. It is explicitly
shown that the $R_\xi$ gauge invariance of the sum of diagrams involving
bosonic fields in the SM also applies to the case of these RS scenarios. An
exact expression for the $h\to\gamma\gamma$ amplitude in terms of the
five-dimensional (5D) gauge-boson and fermion propagators is presented, which
includes the full dependence on the Higgs-boson mass. Closed expressions for
the 5D $W$-boson propagators in the minimal and the custodial RS model are
derived, which are valid to all orders in $v^2/M_\textrm{KK}^2$. In contrast to
the fermion case, the result for the bosonic contributions to the
$h\to\gamma\gamma$ amplitude is insensitive to the details of the localization
of the Higgs profile on or near the IR brane. The various RS predictions for
the rate of the $pp\to h\to\gamma\gamma$ process are compared with the latest
LHC data, and exclusion regions for the RS model parameters are derived.
