Hard to say, speaking objectively I'd probably have to say "Van Halen", or "Jefferson Airplane" (and their various recreations).
This based on fame, longevity, and iconic status.
When going for "Greatest Band" I think the song "White Rabbit" is so iconic that it alone puts "Jefferson Airplane" (also "Jefferson Starship", and "Starship") into a whole differant catagory. Many bands have had famous and long lasting hit songs, but very few that have caught on for as long as that one and arguably served to define an entire genere of music (Psychodelic Rock).
I mention Van Halen, not just because of their success, but also because whenever you get into discussions about the skill of musicians, you can't seriously discuss "who was the greatest guitarist of all time" without bringing Eddie Van Halen into the discussion. A lot of people favor Hendrix over him, but it's very debatable.
In comparison, "The Doors" for example were good, as in really good, but I think some of their fame derives from what happened to Jim Morrison. I also can't think of anything about them that is going to compete with "White Rabbit" or the guitar mastery of "Van Halen" himself. Irregardless of what you might think of the band itself, looking at front men, writers and other things compare them to say "Grace Slick", like her or not, there is no denying her talent is hard to compete with.
This isn't so much about my FAVORITE American band, even though I like both of the ones I mentioned to some degree, I'm trying to look at this as objectively as possible. One of the issues in such a discussion is of course time frame, and part of my arguement about "Jefferson Airplane" and "White Rabbit" is that they did their big stuff back in the 1960s, and "White Rabbit" was their second hit (can't remember the exact date) it has to be close to 50 years old and is still a ridiculously famous song, they even used it for the "Lost Odyssey" TV ads if I remember.