No human beings could not survive without any vegetation on Earth. There are two basic ecological reasons for this: 1. plants provide us with oxygen, 2. plants are at the base of all food chains.
1. Ecologically, plants (flora) and animals (fauna) are interrelated in what is known as the "carbon cycle." The carbon cycle involves the passing of basic chemicals, particularly oxygen and carbon, through a complex cycle involving all life on earth.
Plants in general subsist not only on sunlight, water and soil nutrients, but also on carbon dioxide, which they absorb from the atmosphere. The carbon molecules are stored and used by their metabolism, while the oxygen is released into the atmosphere. This means that forests, for example, are "carbon sinks" which not only store carbon but also cleanse the atmosphere of excessive amounts of carbon dioxide (CO2). This is why both tree planting and preventing deforestation are important steps for dealing with global warming, since they not only help to lower the amount of CO2 in the atmosphere, but also take carbon "out of circulation," as it were.
Animals are the opposite. They breathe in oxygen and exhale CO2, since carbon is a "waste" product where animals are concerned. The animal and vegetable are thus two poles of a causal cycle, each complementary to the other. Under natural conditions, the larger carbon cycle maintains the atmospheric concentration of both gases within the optimal range for the existence of contemporary terrestrial life.
Consequently, eliminating /either/ plants or animals would eventually result in the death of the other. This is because each /require/ the existence of the other in order to continue to live themselves.
The carbon cycle is also an excellent example of the cyclicity, or recursiveness of ecosystem processes at all levels. The same materials are continually recycled through the system, with no loss or waste. Yet this cycling is necessary to the existence of the biosphere as whole.
2. The second reason animal life, including humans, could not continue to exist without vegetation is that vegetation forms the base of all food chains. Food chains are another manner in which all life is interlinked in larger patterns of material and energy exchange. Food chains are also cyclical, and involve the passing of energy and materials from one "trophic" or nutritional level to another.
For example, a grass hopper eats grass. A frog eats a grass hopper. A trout eats a frog. A human eats a trout. At each stage much of the material of the lower stage is directly consumed and digested, but only a portion of the energy (calories) pass from one trophic level to the next (roughly 10 percent).
In general, herbivores or plant eaters are "above" plants, while carnivores or meat eaters are "above" herbivores, or "higher" in the food chain. In general, those species in "lower" trophic levels also exist in greater numbers than those in "higher" levels. This is necessary both because carnivores must eat /several/ herbivores to survive, for example, but also because only a portion of the energy from lower levels is passed up the food chain. Thus, food chains are often represented as a having a pyramidal structure, or even referred to as "food pyramids."
In reality, however, as with most patterns of organization in ecosystems, the food "pyramid" is actually a cycle. This is because plants and animals at all levels, even those at the "top" like humans, also die. In death, they return their bodily substance to the soil, where various "decomposers" (worms, insects, soil bacteria) break this organic substance down and turn it back into soil.
Soil from which plants grow.
So I'm sure you can see that without plants, the entire /food web/, as it is more accurately called, would collapse.
Of course, one could now venture into the realm of science fiction and speculate about mechanical processes or genetic engineering replacing these natural processes through artificial means (such as synthetic food or atmospheric processors). But such technologies are fiction. And given currently available technology, and currently available knowledge of ecosystem functioning, human life is inextricably linked to the existence of vegetation.
Quite simply-without vegetation, no life of any kind more complex than single-celled organisms would exist on this Earth, let alone humans.