Well, the first knives that came to my mind have followed pretty much along the lines of what's been mentioned. Which is to say that I think there are multiples!!!
Perhaps it's better to think of A single knife for a given time frame or style.
The Buck 110 was the very first knife that jumped to my mind. Like most things American, I think it's a design born out of a melting pot. Place an older Bowie-style hunting knife and a French/Spanish locking folder into a blender and you get the Buck 110. But as others have noted, it is more of a recent thing as far as history goes - rather recent. One reason why I think the 110 is so important is that it legitimized the locking folder. It's impossible for me to think about the existence of today's tactical folders without drawing a line back through the Buck 110. It also has achieved icon status in the world of customization and modifications. Sort of like the Chevy small block v8 of knives. Big, bold, mass produced and destined to be modified and hot-rodded.
In terms of fixed blade knives, the Bowie style hunter and the Kabar fighting knives (one big family) are easily THE American fixed blade.
I have a harder time accepting any single slip joint as being THE American pocket knife though as I see them to be more regionally focused. I grew up in the Northeast and grew up seeing more 4 blade campers of various sorts than any others. Stockmen, Congress styles and Barlows all have their place in our history as well. (And all borrowed and deriving from earlier European designs and influences).
In terms of today, I think the case that was made for the Leatherman is very strong. I would add to that the Spyderco folding tactical, which I think is the first folder to use the thumb opener hole and pocket clip.