Full-visible-spectrum white light-emitting diodes (LEDs) with ultrahigh color rendering index (CRI), which are capable of offering a light source that mimics natural sunlight, have great application potentials in healthy lighting, high-resolution displays, and plant growth lighting. However, cyan gap in the 480–520 nm wavelength range is commonly existed in the emission spectra of traditional phosphor-converted white LEDs, and thus it is urgent to develop high-performance cyan-emitting phosphors to close this spectral gap for realizing full-visible-spectrum LED lighting. Herein, we demonstrate a new efficient cyan-green-emitting Ce3+-activated CaY2ZrGaAl3O12 garnet phosphor for near-ultraviolet (near-UV) pumped full-visible-spectrum warm-white LEDs. The CaY2ZrGaAl3O12:Ce3+ cyan-green phosphor has a cubic structure with the space group Ia3‾d and lattice parameters of a = b = c = 12.39645(8) Å and V = 1886.08(6) Å3. Ce3+ doping concentration dependent luminescence properties have been studied, and the highest emission intensity is obtained at 1 mol%. The optimal CaY2ZrGaAl3O12:1%Ce3+ sample shows a broad excitation band in the 300–490 nm wavelength range with peak at 409 nm due to the 4f→5d transition of Ce3+ ions. Upon 409 nm excitation, CaY2ZrGaAl3O12:1%Ce3+ produces a wide cyan-green emission band in the 425–750 nm spectral range with a peak at 514 nm and a full width at half-maximum of 110 nm owing to the 5d→4f transition of Ce3+ ions, and the corresponding CIE color coordinates and internal quantum efficiency are determined to be (0.2802, 0.4856) and 56%, respectively. Furthermore, the resulting CaY2ZrGaAl3O12:1%Ce3+ phosphor exhibits good thermal stability, and its emission intensity at 423 K is about 58% of that at 303 K. Finally, a white LED device is fabricated by employing a 395 nm near-UV chip with the phosphor blend of commercial BaMgAl10O17:Eu2+ blue phosphor, as-prepared CaY2ZrGaAl3O12:1%Ce3+ cyan-green phosphor, commercial (Ba,Sr)2SiO4:Eu2+ green phosphor and commercial CaAlSiN3:Eu2+ red phosphor, which exhibits a bright full-spectrum warm-white light with low color correlated temperature of 3334 K and ultrahigh CRI values (Ra = 97, R9 = 95.3, R12 = 90.1) for 20 mA driving current. This work provides a new perspective in designing luminescent materials for high-color-quality full-visible-spectrum LED lighting.