The CSD has asked the tribunal to assess whether the remarks may have violated rules protecting sporting dignity and decorum.
Tebas made the comments on Friday on social media and in response Real released a statement asking for the 64-year-old's immediate resignation.
Tebas said he was not comparing the ideology of Real or their fan base with those of the Nazi Party but merely the club's propaganda techniques. On Thursday, the CSD said it had raised the issue with the sports tribunal.
The CSD asked the sports tribunal to assess whether Tebas' statements constituted a "potential violation of Article 76.4.b of Law 10/1990 on Sport, which classifies 'notorious and public acts that infringe upon sporting dignity or decorum' as serious infractions."
Sanctions for such infractions include warnings, fines or potential suspension.
LaLiga did not immediately respond to a request for comment.