We demonstrate analytically and numerically that the detection of the spectral response of a single spherical Au nanoantenna allows one to map very small (down to 5·10(-4) RIU) variations of the refractive index of an optically transparent sample. Spectral shift of the dipole local plasmon resonance wavelength of the nanoantenna and the spectral sensitivity of the method developed was estimated by using simple analytical quasi-static model. A pointed scanning probe based on fiber microaxicon with the Au spherical nanoantenna attached to its tip was proposed to realize the RI mapping method. Finite-difference time-domain numerical simulations of the spectral properties of the proposed probe are in good agreement with the theoretical quasi-electrostatic estimations for a radius of the nanoantenna not exceeding the skin depth of Au.