The Argentinian was cited for the incident, which occurred during the Tigers' dramatic 21-20 victory on Sunday.
Club officials reviewed the incident and held and internal disciplinary hearing today.
A club statement said: "Marcos has been banned for eight days, and he will miss the EDF Energy Cup final against the Ospreys, which is a huge disappointment for a young man in his first year as a professional player.
"Marcos is very sorry to have let down the club and the supporters, and would like to apologise to Stade player Remy Martin in respect of the incident."
Ayerza will also appear before an independent disciplinary committee as a result of the citing for alleged punching/striking in contravention of law 10.4 (a).