Rudiger was already substituted when the incident occurred late in extra-time and was subsequently shown a red card as Barcelona won the contest 3-2.
Lucas Vazquez was given a two-match suspension after vehemently protesting the same incident as Rudiger.
The 32-year-old Rudiger underwent knee surgery on Tuesday which will rule him out for the remainder of the season with Real still having five matches to play in LaLiga - he will serve the final game of the ban at the start of next season.
The German defender took to social media on Sunday to apologise for his behaviour towards referee Ricardo de Burgos Bengoetxea.
He wrote: "There's definitely no excuse for my behaviour last night. I'm very sorry for that. Before the final whistle I did a mistake.
"Sorry again to the referee and to everyone I have disappointed last night."
Real Madrid midfielder Jude Bellingham was also sent off after the final whistle, but will not serve a ban after a review of the evidence found an error in the referee's report.
Barcelona's rising star Lamine Yamal has dismissed criticism of his post-match comments following Saturday's Copa del Rey final, insisting that Real Madrid must first defeat him before questioning his confidence.
Carlo Ancelotti's illustrious tenure at Real Madrid is nearing its conclusion, with the Italian set to take charge of Brazil's national team, sources close to the club have confirmed.
Federico Valverde emphasised resilience and determination after Real Madrid's agonising 3-2 loss to Barcelona in the Copa del Rey final.
Barcelona's 3-2 victory in the Copa del Rey final against Real Madrid sparked wild celebrations among the club's supporters.
