Spreading of sessile water droplets on a solid graphite surface has been studied using atomistic molecular dynamics techniques. The system is in the partial wetting regime and a straightforward scheme is applied to extract the contact angle from the atomistic simulations. Even for droplets containing a thousand molecules, the calculated contact angle for water is 83degrees, which is close to the experimental macroscopic value. On addition of ethanol, the contact angle decreases to 30degrees and we approach complete wetting. Time-averaged structural analyses show that the ethanol molecules form a monolayer on top of the solid surface. A few A above the solid surface, the ethanol molecules are preferentially located at the liquid/vapor interface rather than in the bulk of the droplet.