A typical burger meal is definitely processed food because the meat is processed mechanically and mixed from different sources (usually with added fat); the bun is made from refined flour and refined oils; and the fries are thinly-cut potatoes blanched and fried in more refined oils. (Not to mention a whole bunch of chemical additives which are the norm rather than the exception; and "hyperpalatability" tweaks which means the food is formulated to taste unnaturally good.)
You could perhaps argue that a home-made burger meal made using whole cuts of meat, a lean wholemeal bread dough, and potatoes roasted in animal fat could count as a healthy/whole food meal, but it would taste different to the typical hamburger meal we all associate with fast food.
Fast food also has an enormous marketing industry behind it which pushes it in unhealthy volume, with maximum convenience, and at (until recently, at least) affordable prices.