American Football does not meet the criteria of what is sport:
Sport (or, in the United States, sports) is all forms of competitive physical activity which,[1] through casual or organised participation, aim to use, maintain or improve physical ability and provide entertainment to participants.
and
Definition
See also: Game#Definitions
Show Jumping, an equestrian sport
The precise definition of what separates a sport from other leisure activities varies between sources, with no universally agreed definition. The closest to an international agreement on a definition is provided by SportAccord, which is the association for all the largest international sports federations (including association football, athletics, cycling, tennis, equestrian sports and more), and is therefore the de facto representative of international sport.
SportAccord uses the following criteria, determining that a sport should:[1]
have an element of competition
be in no way harmful to any living creature
not rely on equipment provided by a single supplier (excluding proprietary games such as arena football)
not rely on any 'luck' element specifically designed in to the sport