I would argue that this is [edited out: Reductio ad absurdum] a straw man argument. One of the best 'logical' reasons for not eating meat, as opposed to ethical arguments, is the reduction of greenhouse gases and other negative outputs the lower an individual eats on the foodchain. This argument doesn't suggest that an individual starve or die, but instead pushes that individual towards more careful use of the resources available to humanity.
In addition, an ethical argument could be made for not causing pain. Since the vast majority of life on the planet is incapable of experiencing pain, it leaves the vast majority of food that humans eat available for consumption.
So there exist logical arguments against eating meat that do not leave a person with nothing to eat.
Edited because Reductio ad absurdum is not a fallacy, but a straw man is a fallacy brought about by faulty application of Reductia ad absurdum.