It's generally the "fate" of every rapidly increasing population such as our own that they eventually reach a point where they deplete/don't have enough resources, get stressed from being too many lumped together, easily spread diseases through the population and other negative effects so that the numbers drop. Sometimes, population growth stops increasing and "stabilizes" and other times, they crash leaving only a relatively few survivors compared to population height numbers.
The crash scenario is most easily illustrated for us if you imagine billions of people eating all the food there is to eat .. and then starve to death.
I think there's a fair chance we're heading for such a scenarion and shifting diet to plants may increase the amount of time it takes until we get there. That alone wouldn't solve the problem, though, we can still deplete edible plant resources. What we need is to live sustainably, but what's sustainable with a world population of 7 billion people is not necessarily sustainable in a world with 14 million people.
In other words, veganism is not a solution to that problem. You'd need something more extravagant like a way to control the world population and keep it from increasing to the point where we're too many .. for example by controlling how many kids families have. On a global scale, that's a near impossible political task. Divided nations and all that aside, we're not exactly bees working in the best interests of the collective hive.
edit :
Well, Erica is
