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In contrast to an infinite degree of nonlocality, we demonstrate that vortex solitons in nonlinear media under competing self-focusing cubic and self-defocusing quintic nonlocal nonlinearities can be stabilized with a finite degree of nonlocality. Stable vortex solitons in the upper branch, bifurcated from the competing cubic-quintic nonlinearities, are found to be supported when the original double-ring refractive index change is transferred into a single-ring configuration due to the balance between diffusive nonlocality and defocusing quintic nonlinearity. The dynamics and stabilities of the vortex solitons are studied analytically and numerically. [ABSTRACT FROM AUTHOR]