Yes, it's natural. In fact, we've been eating meat for so long it is natural.
Weave - Your argument about building up a tolerance to meat rests upon sloppy foundations. Most of your evidence is anecdotal at best. In fact, to your hypothesis about it being a biological element, I believe it to be psychological. The rejection is a result of your mind telling you it's wrong to eat meat thus the body responds accordingly.
Being omnivorous, I think, gives humans a great advantage in terms of survivability in every possible environment as D333 pointed out. When you look at the species that are the least affected by a change in the ecosystem, most if not all are omnivorous. Roaches for example, can eat absolutely anything and they've outlived almost everything.
Then there's the question of overpopulation. Simply putting in on the back of a single factor is highly simplistic. On one hand, the industrial production brought food stores to an incredible high, which ensured a bigger population. Then again, this holds for the western countries, as southern and eastern populations also saw a rise in their population without the incredible technological advancement to back it up. Famine still swept those regions, yet they outnumber the westerners today.
The humans also don't have any predator to keep its population in check. In fact, with our ability to cure diseases, there is literally nothing to properly balance out our population. The advances in medicine contribute to the greater lifespan of humans in our modern world.
So meat being a factor of overpopulation? I don't think so.