The 24-year-old Harlequins prop was cited for making contact with Michael Paterson's eye during the Premiership defeat by Northampton on Saturday, and will now miss England's matches against Argentina and Australia at Twickenham on November 11 and 18 respectively.
Sinckler, who will be free to play for England when they face Samoa on November 25, has apologised for the incident, saying it was not his intention to hurt his opponent.
"It was a genuine mistake and an act of recklessness on my part," he said. "I accept the outcome of the hearing and wanted to go on record to say I am sorry that I have let my team-mates down.
"More importantly, I feel terrible that anyone would think I would deliberately gouge an opponent. That was never my intention."
The ban also means that Sinckler will be forced to miss seven Harlequins matches, including their European Cup fixtures against La Rochelle and Wasps.