Simeone was sent to the stands during the first half of the clash at the Vicente Calderon after aggressively remonstrating with the fourth official, who he then tapped on the back of the head.
The Argentinian's ban has been made up of four different punishments and includes a combined fine of 4,805 euros (£3,800). Atletico have also been fined 2,800 euros (£2,230).
Simeone has 10 days to appeal the sanctions, which apply to any RFEF competition.
The four sanctions for Simeone included a four-match ban for twice tapping the official's head and a two-match suspension for the protests which led to his sending off.
Simeone was also given two further bans of one match each for sarcastically applauding the official's decision and for remaining in the stands instead of going to the changing rooms after his dismissal.
Simeone, who has apologised for his actions in Friday's game which Atletico won 1-0 thanks to Mario Mandzukic's early goal to lift the Supercopa 2-1 on aggregate, was also sent off in his side's Champions League final defeat to Real in May.
The 44-year-old was dismissed following an ugly on-field spat with Real defender Raphael Varane just before the end of the 4-1 extra-time defeat, and was subsequently handed a one-match European ban by UEFA.
Atletico assistant coach German Burgos will take charge in Simeone's absence, with the Primera Division champions in action on Monday night against local rivals Rayo Vallecano in their season opener.