Primordial black holes (PBHs), possibly formed from the collapse of early universe perturbations, will evaporate via Hawking radiation with a lifetime comparable to the age of the universe, if their mass is $O({10}^{14})$ g. Such black holes can contribute to the observed gamma-ray fluxes in the MeV and GeV range. Using the observed extragalactic gamma-ray background (EGRB) from the Large Area Telescope, the , and the telescopes that cover gamma-ray energies from 0.5 MeV to 1 TeV, we evaluate limits on the abundance of PBHs with masses of ${10}^{14}$ to ${10}^{17}$ g. We study both monochromatic and extended mass distributions of PBHs. To model the EGRB spectrum, we calculate the contribution from extragalactic sources including blazars, star-forming galaxies and radio galaxies and also account for ultra-high-energy cosmic rays that produce gamma rays when interacting with the infrared background. Our EGRB modeling uses information from the gamma-ray point sources catalog, from observations at X-rays, the visible spectrum, the infrared and radio waves, and also accounts for modeling uncertainties and variations on the properties within each class of these sources. Moreover, we use recent work on the modeling of the PBHs’ gamma-ray emission, that includes the direct Hawking radiation, gamma rays produced in the hadronization and decay of unstable particles, final state radiation and gamma rays from pair annihilations in the interstellar medium. As the contribution of final state radiation and the annihilation of positrons enhances the low-energy part of the produced gamma-ray spectra from PBHs, we find that the EGRB observations can set the tightest limits on their abundance among all indirect dark matter probes, within the mass range of interest. PBHs with masses $\simeq {10}^{14}$ g cannot contribute more than $O({10}^{-10})$ of the observed dark matter. Finally, we discuss the impact of future observations from AMEGO-X and e-ASTROGAM, finding that these future MeV-scale telescopes can improve the current limits on PBHs of masses greater than $5\times {10}^{14}$ g by up to a factor of one hundred.