Smith, 30, was hit in the neck by a 92mph bouncer from Jofra Archer on day three of the second Test, but passed an initial concussion test and returned to continue his innings.
Coach Justin Langer confirmed his absence after Smith sat out Australia’s training session on Tuesday.
Smith has scored 142, 144 and 92 in his three innings so far in the series.
It has not been confirmed who will replace Smith in the Australia squad for the third Test at Headingley.
Smith was replaced by Marnus Labuschagne in the second Test at Lord’s, with Labuschagne becoming Test Cricket’s first concussion substitute.
He made 59 as Australia survived a tense final evening to claim a draw and preserve their 1-0 series lead.