The defender was left out of the squad for Sunday's 1-1 league draw at Club Tijuana after images he shared from his son's 12th birthday party on social media went viral on Jan. 6.
Cruz Azul had been considering Dominguez's future at the club following the incident, sources told ESPN.
The celebrations saw guests playing with fake weapons, wearing Cruz Azul jerseys and caps with the initials "JGL," in reference to Joaquin Guzman Loera 'Chapo,' a Mexican former drug lord who is serving life imprisonment in the United States.
The party took place hours after 19 suspected gang members and 10 military personnel were killed in a wave of violence in the Mexican state of Sinaloa following operation on Thursday by the Mexican government that ended with the arrest of Ovidio Guzman, the son of 'Chapo.'
The photographs of Dominguez's party that circulated in social media were considered by Cruz Azul fans to be distasteful and inappropriate, with some supporters calling for the player to be released from his contract.
In a joint statement Tuesday, Liga MX and Cruz Azul condemned the "unfortunate and unacceptable events."
It added that Cruz Azul will provide support, training and awareness to the player and assured that Dominguez would release a public message on Tuesday.
Dominguez, 35, later apologised in a video statement, saying that "it was a mistake" and he accepts the "distress that the images caused to the Mexican society".
"I want to offer a sincere apology to the society, fans, my teammates, fellow players, the clubs, Liga MX and the Mexican Soccer Federation for these unfortunate events.
"I recognise that these images are not what Mexico needs. I have committed myself from now on to become a benchmark that promotes the values of Liga MX and my club," Dominguez said.
The defender reacted positively during the talks with club executives about the incident. It remains to be seen if the club or Liga MX will announce further sanctions for the player.
Dominguez, who holds the all-time record for most appearances for Cruz Azul, has represented Mexico on 24 occasions, his last coming in June. He did not make Mexico's World Cup squad.