The afterglow emission from gamma-ray bursts (GRBs) is believed to originate from a relativistic blast wave driven into the circumburst medium. Although the afterglow emission from radio up to X-ray frequencies is thought to originate from synchrotron radiation emitted by relativistic, non-thermal electrons accelerated by the blast wave, the origin of the emission at high energies (HE; $gtrsim$~GeV) remains uncertain. The recent detection of sub-TeV emission from GRB~190114C by MAGIC raises further debate on what powers the very high-energy (VHE; $gtrsim 300$GeV) emission. Here, we explore the inverse Compton scenario as a candidate for the HE and VHE emissions, considering two sources of seed photons for scattering: synchrotron photons from the blast wave (synchrotron self-Compton or SSC) and isotropic photon fields external to the blast wave (external Compton). For each case, we compute the multi-wavelength afterglow spectra and light curves. We find that SSC will dominate particle cooling and the GeV emission, unless a dense ambient infrared photon field, typical of star-forming regions, is present. Additionally, considering the extragalactic background light attenuation, we discuss the detectability of VHE afterglows by existing and future gamma-ray instruments for a wide range of model parameters. Studying GRB~190114C, we find that its afterglow emission in the fermi-LAT band is synchrotron-dominated.The late-time fermi-LAT measurement (i.e., $tsim 10^4$~s), and the MAGIC observation also set an upper limit on the energy density of a putative external infrared photon field (i.e. $lesssim 3times 10^{-9},{rm erg,cm^{-3}}$), making the inverse Compton dominant in the sub-TeV energies.
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