We experimentally investigated the coalescence of two sessile microdrops on rigid surfaces with diverse wettability (macroscopic apparent water contact angles of theta(app) approximate to 13-110 degrees) and on hydrophobic surfaces (theta(app) approximate to 110-124 degrees) with very different stiffness properties (Young's moduli of E approximate to 1.1 MPa to 130 GPa). We show that the coalescence contains two fast regimes, in which a liquid meniscus bridging the parent droplets rapidly grows, forming a hemi-ellipsoidal droplet, and a slow regime, in which the merged hemi-ellipsoidal droplet relaxes to the equilibrium hemispherical cap. Whereas the fast bridging regimes last less than 2 ms and are almost independent of surface wettability and stiffness, the relaxation regime, which was only observed on sufficiently hydrophobic and rigid surfaces with low wetting hysteresis, continues for a few tens to several hundreds of milliseconds depending on surface properties. We further demonstrate that the slow droplet relaxation can be described neither by the bulk hydrodynamics nor by a microscopic model concerning liquid evaporation near the droplet edge, but by the molecular kinetic theory for the motion of the three-phase contact line.