We demonstrate the existence of various types of gap localized modes, including one- and two-dimensional (1D and 2D) single solitons and soliton clusters, as well as the corresponding vortex modes in optical media with saturable Kerr nonlinearity and fractional diffraction. We find that soliton clusters with different number of peaks can be stable in these media. The 1D and 2D localized modes existing at the center of the first and second band gaps are stable, whereas the ones in the peripheries are unstable. In addition, the vortex modes with different number of peaks and vorticity number m= 1 are found to be stable, while the ones with m≥ 2 are unstable. The stability of these localized modes is investigated by using the linear stability analysis and is confirmed by the numerical simulation of their dynamical propagation. The obtained results may enrich the understanding of gap solitons and their complexes in media with saturable nonlinearity and fractional diffraction, and may find potential applications in optical information processing and other related fields. © 2022, The Author(s), under exclusive licence to Springer Nature B.V.