The New York City Board of Health on Thursday approved Mayor Michael Bloomberg’s controversial proposal to ban the sale of large sugary drinks in restaurants and other venues, a sweeping initiative that backers hope will reduce obesity and critics decry as government run amok.

Beginning March 12, the city will prohibit restaurants, mobile food carts, delis and concessions at movie theaters, stadiums or arenas from selling sugary drinks in cups or containers larger than 16 ounces. The city will begin fining sellers for violating...