Reports on Thursday suggested Luiz struck Ceballos in the nose and drew blood after the Spanish midfielder caught his teammate with a rash challenge in training last Friday.
"I don't like the fact that that incident comes out at all," Arteta told reporters Friday, courtesy BBC Sport.
"I will find out where it is coming from and if that is the case, that goes completely against what I expect from each other, the privacy and the confidentiality that we need, and there will be consequences."
Arteta called the incident "nothing" and added, "Training is very competitive and issues happen a lot of times. Those things get resolved immediately within the team and there is not much to say.
"There is no problem at all," Arteta added.
Ceballos shared the initial report on Twitter on Thursday evening, calling the speculation "FAKE."
The incident marks the second time this season Ceballos has been involved in a feud with a Gunners teammate.
Ceballos, who is on loan from Real Madrid for the second successive season, was seen berating striker Eddie Nketiah during warm-ups ahead of a Sept. 12 victory at Fulham.