The fundamental reason humans could not survive without vegetation is that vegetation releases oxygen. Plant life intakes carbon dioxide and releases it as oxygen through the process of photosynthesis. This is the sole reason other complex life forms have evolved on Earth. Without vegetation, humans and most animals could not breathe the air, much less evolved in the first place. Therefore, vegetation is absolutely essential to the survival of humans.
Before plants spawned into existence, the atmosphere was carbon dioxide dominant. Plant life then came into existence as there was a niche to be fulfilled- this niche being the intake and subsistence upon the vast amounts of carbon dioxide. Once plants began sucking in carbon dioxide and releasing oxygen, it allowed for the evolution of more complex life forms, including humans. Now humans would not evolve for millions of years, but this increase in oxygen made there evolution possible. Organisms began breathing in oxygen and evolving with more complex structures and ways of life. Eventually, humans came about by subsisting on the plant life and animal life that previously used vegetation to survive.
Not only is our existence dependant on vegetation, the existence of every mammal, insect, reptile, amphibian, and every living creature on the planet are dependent on vegetation. It is an essential element in our ecosystem. All life fulfills a niche, and if vegetation were to become extinct, all other life would follow suit because there are too many creatures that use plant life to fulfill their niche. Even if an animal does not directly feed and subsist off of pure vegetation, the creature it uses for subsistence most likely relies on vegetation. And if that animal does not rely on vegetation, the next animal in the hierarchy of the food web would, and if not that one, somewhere along line of prey and predator a creature must subsist on vegetation. It is the way complex life forms have evolved. There has to be a primordial existence that all other life forms can evolve from. And vegetation is one of these primordial pieces of evolution.
Evolution is not an accident. It occurs based on other occurrences, a perpetual changing and continuous formation of life. It is dependent upon itself and vegetation is an extremely significant part of the symbiosis that exists between all living creatures on the planet. Vegetation feeds upon the nutrients of the soil, cattle and other animals then feed on the vegetation, and predators then feed on the animals feeding on the vegetation, while feces discharged from the animals replenishes the nutrients in the soil. It is a co-dependant cycle of life. If vegetation were to suddenly vanish, all herbivores would become extinct due to lack of food, and all carnivores would then die off for the same reason, not to mention that the supply of oxygen would dissipate, rendering the air unbreathable.
Another important aspect of vegetation regarding humans is that we have evolved as omnivores. We feed on meat as well as plant life. Without vegetation, humans would become devoid of the essential nutrients required for survival. And once, all other animal life dies off, assuming that humans would be the last to become extinct, which probably would not be the case, there would be absolutely no food for humans to survive on.
It is blatantly obvious why a world devoid of vegetation would render humans extinct, not to mention that humans would have never existed in the first place without vegetation. It is integral to the existence of life in general. Next to water, I would argue that it is the most important piece of the evolution of life. It is life itself, and all other forms of life depend on it and need it. It is an essential part of our evolution and an necessary component for the continuation of our existence.