Ancelotti claimed his first managerial victory at Camp Nou on Monday, as goals from David Alaba and Lucas Vazquez secured a 2-1 victory.
Struggling Barca boss Koeman was then subjected to unsavoury scenes as he left the ground, with the Dutchman's vehicle surrounded by fans as he attempted to depart.
Barca released a statement to condemn those actions and in his news conference on Tuesday, Koeman suggested the incident reflected wider problems in society.
Ancelotti, who Koeman also revealed offered his support, echoed his counterpart's sentiment.
"It is a lack of respect for the person, not for the coach. Koeman is not a coach, he is a person who works as a coach," Ancelotti told the media ahead of Madrid's LaLiga meeting with Osasuna on Wednesday.
"It is not a problem of football, but of society, of rudeness.
"A friend who is also a coach told me that the best life for a coach is when there are no games. It is a beautiful life, we like to train. Criticism is normal, but disrespect is annoying on a personal level.
"When you are a child they teach you, at least to me, if you want to be respected you have to [show] respect."
Despite the victory, some sections of Madrid's support were seemingly unhappy with how Los Blancos played at Camp Nou.
Barca shaded the possession (51.6 per cent) and had more shots (12) than Madrid's 10, though the hosts only managed to get two on target - one of those being Sergio Aguero's last-gasp consolation - and were constantly exposed on the counter-attack.
"I understand the criticism because everyone can have an opinion," reflected Ancelotti.
"I said that we had not played perfectly, but that we played a smart game, which was what we had asked for."
Ancelotti has not lost as Madrid coach against Osasuna in all competitions (W3 D1), though Thursday's match will not be simple, with Jagoba Arrasate's side sitting in sixth, just two points back from Los Blancos.
"Osasuna are a good team that play intense football, that defend well," Ancelotti said. "They are dangerous and their [league position] speaks clearly. It will be a difficult game."