I hate this argument.
Both sports are called "football" because they are played ON FOOT, as opposed to being played on horseback such as the more patrician sports like polo. Manipulation of the ball with your foot has nothing to do with the name.
Sir, you reasoning is the argumentative equivalent of a fart. Funny, but about as close to shit as it gets without literally being it.
Let's recap. Baseball = football. Basketball = football. Golf = football. Tennis = football. Volleyball = football. Handball = football.
That would would be a very pointless distinction, now wouldn't it. Games played horseback: polo. Games played on foot: just about every other freaking ball game besides some waterpolo and other obscurities.