No, human populations would not fall.
In fact, human rights activists and environmentalists often support vegetarianism because it
reduces the use of pasture land and pollution (
see here). Around 50% of soy and corn crops go to feed animals, around 8% of useable water is given to food animals, and at least here in Nebraska I observe huge amounts of pasture used to sustain grazing cattle. Animals are remarkably inefficient machines for converting grains into meat, see
here for the amount of grain goes into each kg of edible meat. Why is all that food being fed to animals when 1/3rd of the planet is undernourished?
From a resource usage point of view, I don't think you can really make an argument for animal agriculture.