We determine contact angles of micrometer-sized NaCl-water droplets on superhydrophobic surfaces by analyzing their lowest-order axisymmetric vibrational resonances driven by vertical oscillations of the surface. Fluorescence spectra of the dye-doped droplets excited by laser light feature whispering-gallery modes (WGMs) whose spectral widths depend on the droplet vibration amplitude, thus enabling precise measurements of the droplet mechanical resonant frequency. Following droplet size determination by WGM mode-matching, we calculate the contact angles from the dependence of the measured mechanical resonant frequency on the droplet size for two surfaces with different superhydrophobicity levels, and find a good correlation with the values measured by direct imaging of millimeter-sized droplets.