Actually, Americans say they speak English because no viable alternative has ever arisen.
Saying we speak American gets us in even deeper water than calling it English does. Denizens of North AMERICA, Central AMERICA, and South AMERICA all are Americans.
The Hispanic speakers call us estadounidense. Translating that, UnitedStatesese is a harsh sounding, cumbersome name for the language of the UnitedStatesians.
USish - ISH is right!
Maybe we need to start calling our version USese, the language of the USians. "My fellow USians," (Revised Richard M. Nixon speech introduction when addressing the "American" (US only) public)
Or better yet, maybe we should invite the English speaking Canadians in and label our language Nor'Amish. Has a nice ring to my ear.