We develop a microscopic theory of the linear electromagnetic response of bilayer excitonic insulators relevant to electron-hole double-layer systems. Using a self-consistent Hartree-Fock description of the excitonic ground state and time-dependent Hartree-Fock for its dynamics, we compute the collective mode spectrum and the full first-order response to layer-symmetric (charge) and layer-antisymmetric (exciton) gauge fields. At zero magnetic field, we find that two gapped plasmon modes dominate the long-wavelength charge response, while the exciton channel is governed by a linearly dispersing phase (Goldstone) mode. From the Goldstone-dominated kernel, we derive London-like equations for the exciton condensate. The nondissipative acceleration under a layer-antisymmetric electric field provides a direct signature of exciton superfluidity; in contrast, a normal exciton fluid shows a Drude-like dissipative response. In a perpendicular magnetic field, the Goldstone mode develops a magnetic-roton minimum that signals an instability toward a finite-momentum stripe-ordered excitonic insulator. In addition, the field couples charge and exciton motion, giving rise to dipole and inverse dipole Hall effects in which a charge (exciton) bias induces a transverse exciton (charge) current. In the condensate, these mixed Hall conductances approach finite values in the ω→0 linear-response limit. Our findings provide concrete targets for microwave and transport probes of bilayer exciton superfluidity.