A gradient-index optical fiber lens is proposed and fabricated on the tip of a single-mode fiber using focused ion beam milling. Second-order effective medium theory is used to design a gradual change in the fill factor, which ensures a parabolic effective refractive index distribution. The proposed fiber lens design is simulated via the three-dimensional finite-difference time-domain method, and demonstrated through confocal optical measurements. At a wavelength of 1550 nm, the fabricated lenses focus a 10.4 μm mode field diameter exiting the fiber into spot sizes between 3-5 μm, located 4-6 μm away from the fiber tip. Direct coupling into a silicon-on-insulator chip is also demonstrated, where the fabricated gradient-index lens has a coupling efficiency comparable to a commercial lensed fiber.