The major challenges in traditional color phase hologram generation are the time-consuming iterative procedure and aberration caused by different wavelengths in color holographic display. Based on the original non-iterative phase hologram generation method-optimized random phase (ORAP), combined with the physical limitations of color holographic display, this paper proposes a full-support optimized random phase (FS-ORAP) method for non-iterative color phase hologram generation. FS-ORAP breaks through the limitation of the original ORAP method in the fixed support constraint of the target amplitude in the spatial domain, the full support constraint can be used to generate phase holograms of target amplitudes with arbitrary support size, which fits well with the generation mode of the three-color channel of the color phase hologram. In addition, the color aberration of the reconstructed image is eliminated by scaling the size of the three-color component. At the same time, FS-ORAP is used for the non-iterative fast generation of three-color channel holograms, which can greatly improve the generation speed of color phase holograms and can be adapted to various color holographic display techniques. Experimental results verify the feasibility of our proposed method.