Suddenly, I envision warrior plants with swords and shields, doing battle with animals and other plants until only one plant is left.
There are too many things wrong with this "argument" to list, but here's something that most idiots should be able to comprehend.
The "fittest" are those organisms that survive in their environments and reproduce, nothing more. Cactii are not fit to live in freezing climates, fish are not fit to live on dry land, and humans are not fit to live under water.
Further, species depend on other species for their survival. For example, plants require carbon dioxide, which is produced by animals. Animals require oxygen, which is produced by plants. One could not survive without the other.
Now for something a little less simple, but no less true...
You depend on certain types bacteria to survive. When your immune system becomes stronger, it's because beneficial bacteria has survived a particular threat, then gone on to reproduce, while the bacteria that didn't survive the threat dies. If you sanitize your surroundings too much, your immune system actually becomes weaker. Because these threats are being killed before they come into contact with this beneficial bacteria, the beneficial bacteria in your body isn't able to ward them off when you do become exposed to them.
There is no way one individual organism -- or even one species -- could survive without other species, and I repeat: The "fittest" are simply those organisms that survive in their environments and reproduce.