Video posted by Spanish media on Sunday showed Cadiz fans yelling insults at Rudiger after he went to the stands to give away his shirt to someone who appeared to be a Madrid supporter. Objects were also thrown at the German defender, who is Black.
Rudiger exchanged a few words with some of the fans yelling at him before giving his shirt away and saluting some of the supporters. Two people who appeared to be members of Madrid's staff came over to lead the player away from the fans.
The incident happened after Madrid's 2-0 win against relegation-threatened Cadiz.
Spanish football has been marred by racism this season, with Rudiger's teammate Vinicius Junior having been targeted during several matches.
Earlier this month, the Brazil international testified against a Mallorca fan who was accused of racially abusing him during a match. The same fan was also accused of insulting Villarreal midfielder Samu Chukwueze in another league match.
The Spanish league has filed several formal complaints to authorities over insults against Vinicius, with some of them having already being shelved. The league also recently increased the number of monitors at Madrid matches to identify perpetrators.
Valladolid recently suspended 12 season-ticket holders while investigating alleged verbal abuse of Vinicius.
The first trial against a fan who racially abused a player in Spain is expected to begin this year following remarks by an Espanyol supporter against Athletic Club forward Inaki Williams a few years ago.