When the vapor pressure exerted by the gas equals that exerted by the liquid, do their densities equal?
V, you are mixing terms a little.
Vapor pressure is a quantity indirectly describing the amount of a substance must be in the gas phase for the substance to be in equilibrium between the liquid and gaseous states. If the partial pressure of the gas (say water vapor) is equal to the vapor pressure of the liquid (must be the same substance, so water) than the liquid will have no net flow either from the gas phase to the liquid phase or vice versa.
There is no need for the substances to have the same density. This is only true for super-critical fluids. Super-critical fluids exist once you have raised the temperature above a critical point after which there is no phase transition between liquid or gas.