We address the existence and stability of fundamental, single-charged vortex, and double-charged vortex gap solitons in two-dimensional quasiperiodic photonic lattices imprinted in a Kerr-type medium. Fundamental and vortex gap solitons can bifurcate from linear localized states or their combination supported by quasiperiodic lattices for both defocusing and focusing nonlinearities. We find that the three types of solitons mentioned above are stable in the entire existence domain for defocusing nonlinearities, and that they can also be stable at a lower power level for focusing nonlinearities. At higher power, unstable solitons are characterized by a ring-shaped symmetry-breaking distribution, and the unique spot profile formed is repeatedly observed with changes in propagation distance.