We report and demonstrate a highly-sensitive refractive index (RI) sensor based on a linear-cavity dual-wavelength erbium-doped fiber laser (DWEDFL). The optical spectrum of the laser varies as the external environmental RI changes from 1.3 to 1.335. The DWEDFL has a linear-cavity configuration with two fiber Bragg gratings (FBGs) with central wavelengths < 1 nm apart. Since both FBGs share the same EDF gain medium, gain competition occurs in the cavity. Optical loss of one wavelength can be introduced by immersing the sensing component, a 15 mm micro-fiber (MF), in a solution under test. Experimental results demonstrate a high sensitivity of -231.1 dB/RIU (refractive index unit) and 42.6 dB/RIU in the range from 1.300 to 1.335. The relative power change at the two FBG wavelengths reveals a higher sensitivity of -273.7 dB/RIU with better stability due to reduced light source jitter and external perturbation. Due to its high sensitivity and simple structure, the dual wavelengths gain competition RI sensor has potential applications in chemical and biochemical sensing fields.