The first-half incident had gone unpunished during Friday's Premiership match, but he was later cited.
Later in the game, Clark, 27, was yellow-carded for bundling a Warriors player into touch off the ball.
He will miss league matches against Newcastle and Leicester, and the Champions Cup game against Leinster.
"The panel found this was not a premeditated act but an intentional strike to the head of an opponent that merited a mid-range entry point of three weeks," an RFU statement read.
"The player's remorse before the panel was genuine and heartfelt."
Northampton flanker Calum Clark has been banned for three weeks after pleading guilty to striking Worcester's Donncha O'Callaghan with his elbow.