Under ambient conditions, a water meniscus generally forms between a nanoscale atomic force microscope tip and a hydrophilic surface. Using a lattice gas model for water and thermodynamic integration methods, we calculate the capillary force due to the water meniscus for both hydrophobic and hydrophilic tips at various humidities. As humidity rises, the pull-off force rapidly reaches a plateau value for a hydrophobic tip but monotonically increases for a weakly hydrophilic tip. For a strongly hydrophilic tip, the force increases at low humidities (<30%) and then decreases. We show that mean-field density functional theory reproduces the simulated pull-off force very well.