Mr Doucet responded by saying his right-wing predecessor as mayor introduced the same measure during the pandemic last year.
Fish and eggs remain on school menus in Lyon, and Mr Doucet said menus would be balanced for all children.
Nutritionists say a vegetarian diet is safe for children, but advise taking extra care to ensure sufficient amounts of protein, iron and other minerals are included.
Lyonnaise cuisine is world renowned, with a particular emphasis on meat and offal.
But there are signs of tastes changing. Sales of meat-free products are on the rise, and a 2018 law made it compulsory for schools to make one day vegetarian. A vegan restaurant recently made history in France by winning a coveted Michelin star.
Mr Doucet maintains the measure is solely as a result of the pandemic, but has been unafraid to challenge French traditions. He has previously called the Tour de France bike race “macho and polluting”.