There are some amongst us that would argue the concept was a genuinely American one...granted the Germans coined the term Sturmgeweher and the weapons were designed for two very different roles, but when it came to actual battlefield usage, there really wasn't much difference...
It can be argued that the .30 Carbine round was one of the first assault-type medium cartridge, but in reality it was closer to a pistol cartridge than rifle cartridge in performance.
Howerver, the M1 Carbine that fired it was incapapble of full-auto fire. The selective fire M2 was not introduced until 1944, around the same time at the STg.
Therefore, in my mind at least, it was not the first assault rifle.