I write this contribution partly as an answer to the question raised by Juran's creative contribution. If such a ban was established, this would definitely result in a huge increase in animal farming. Switching to a plant-based diet is a good solution, but animal farming is even worse than hunting/fishing in my opinion. Firstly from an ethical standpoint - farmed animals have the worst "life" one could imagine - they are forcefully bred into existence, kept mostly in cages, offsprings are separated from their moms, the animals are often kept in very bad conditions and eventually, all of them are exploited for the products of their bodies one way or another, they are treated and killed usually very cruelly. Also, psychological health of people who work in slaughterhouses is usually affected.
In the wild animals at least have freedom and chances to survive. They are used to the "kill/get killed" survival mode, it's part of their life. Farmed animals, on the contrary, are like pets intended for abuse. Therefore sustainable use of wild resources, including hunting and fishing seems a better option in that regard. On top of the ethical concerns, animal farming also contributes a great deal to global warming and environmental pollution. Some experts speculate that meat consumption might become illegal in the future because of this ecological reason.
While an assumption that a wholesome diet has to include animal products is still mostly prevailing in society, more and more evidence from various scientific studies is piling up to debunk this. Plant-based diet has been proven to help prevent many serious health problems, including cardiovascular diseases, diabetes and even cancer. It's true that a few vitamins and microelements are difficult to get from a plant-only diet, but using supplements is a rather negligible price to pay compared to the disadvantages of animal farming. An alternative option is occasionally supplementing plant-based diet with animal products from sustainable fishing/hunting. But essentially I don't think we even need that, once plant-based diet would become prevailing in society, the "lack of nutrients" problem wouldn't be a problem anymore, cause mass-production would be oriented towards necessary fortification.