Menu = Complete Meal
Speisekarte=A presentation of food and beverage offerings
From the discussions below, I would conclude that "Menü" is what we would call a "set meal" at the restaurant, such as a "Happy Meal" at McDonald's which would include the fries and coke, whereas a "Speisekarte" is what we would call a "menu", the card or booklet or paper that lists all the food etc. Am I right?
Speisekarte is the list of everything the restaurant sells to eat or drink. It's the physical paper or cardboard if you want. A "Menü" is a special and specific combination of things that makes a complete meal (appetizer + main course + dessert OR main course + dessert for instance) you can get for (usually) a cheaper price than the sum of the prices of the individual items. In France as far as I know you typically have these kind of deals at lunch, when people are more likely to only order a main course.
Menu is feminine, and I get that's why 'eine' is there, but I thought this context was accusatory. If I'm eating an apple it's "Ich esse einEN Apfel" but if I'm reading a menu it's "Ich esse einE Speisekarte"? If it's accusatory, it would say ein Speisakarte, right? What makes this not accusatory?