you mean football, a sport played by kicking a round ball virtually all the time
This is where things get a bit muddled.
It is true that water polo was once called "Football-On-The-Water but that was too long of a name and dangerous to boot due to the risk of drowning. It did not become Handball, although it must be considered that the name was already taken by a different kind of ball-centric sporting event which is the issue here as well.
Then there's regular polo. It is definitely not called Malletball-Played-By-Rich-Guys-On-Horseback, though it could be.
And hockey, another sport whose all-to-familiar theme is to launch an object into a net, is not referred to as Stickpuck-On-Ice even though it would so qualify.
Lacrosse, another run around and get the ball in the net sport is not called Longstickball or Stickball for short. The latter actually a game popularized on the streets of New York and other eastern US cities. This game more closely resembles baseball than cricket, the actual sport that morphed into baseball in function and name within the western hemisphere, at least initially. Plagiarizing that name would also have caused confusion if there had been American cricket to go along with the rest of the world's cricket. Besides, we already had Buddy Holly and the Crickets over here beginning well over half a century ago and they utilized musical instruments, not sporting props -- although that's not to say that even more confusion and debate would not have ensued if an unprecedented THREE distinctly different disciplines used the common name of cricket. No doubt it would have.
That brings us to which fans should call their particular sports "football." Let's ponder this for a moment. Football, aka soccer in the USA, indeed exclusively uses feet to propel a ball towards a net except for cheating purposes and also by the goalies and refs who can use their hands, too. And it's not called Feetball either although that would be the more appropriately descriptive name of the two because it requires pedal ambidexterity. It could be subtitled as Headball-Once-In-A-While or even Athletes-In-Short-Shorts-Who-Get-Wildly-Indignant-Whenever-A-Piece-Of-Paper-Is-Thrust-In-Their-Faceball. Way too verbose. No argument from me there. Like it or not, let's also keep in mind that no other sport shares the name soccer, so it would be exclusive to the sport. Football or Soccer? At least either would conveniently distill it all down to just a single immediately recognizable name. Like Beyonce, Elvis, or Pele.
As for the American version of football, the much too verbose Chuck, Run, and/or Catch-An-Oblong-Pointyended-Laced-Object-Over-A-Goaline-Or-Close-To-It-Then-Kick-It-Through-Upright-Pipes-For-Good-Measure pretty much describes the sport by its name alone. Since the punctuation to a score or the score itself is done with the foot. Well, you know what that comes down to calling it.
Solution: My recommendation would be to modernize the game of football (aka soccer in the USA) through changing its name to Pedosphere since that is also the surface that it is not-so-coincidentally played upon. Honk your horn if you agree.