Melee comes from French, to fight, and is pronounced by French rules, with accents it is mêlée, which in English the e-circonflex would be more like "e" in Greg, and ée would be "ay" as in "gray".
Most commonly English speakers butcher it however they like, so no, there is not one correct way to fail to pronounce the French word. It's a potayto-potahtoh thing.
Most commonly I hear "Meh-lay", from US video-game users who use the term to refer to close-ranged weapons like knives, in first person shooters.