WARNING: The following article contains spoilers for the Justice League Dark story "Dangerous Things" by Ram V, Xermanico, Romulo Fajardo Jr., Rob Leigh, Andrea Shea, and Alex R. Carr, featured in Justice League #60, available now.
In the latest Justice League Dark backup story from Justice League #60, the Justice League and the Justice League Dark discuss Etrigan's involvement in helping defeat the latest magical threat. As they discuss whether Etrigan can be trusted to rejoin the League's magical counterpart, Batman interjects saying that it should be Zatanna's decision as she should be new the leader of Justice League Dark. Batman's words rally the rest of the heroes around Zatanna but, Batman isn't finished with the issue just yet.
Batman takes Etrigan to the side and confronts him about whatever his true intentions may be. Batman threatens Etrigan, telling the demon that the only reason he stood up for him and the Justice League Dark was because of Zatanna. Specifically, because he feels he owes her father -- Zatara.
When Zatara came to Gotham in his early years he befriended a young Thomas Wayne. Zatara's nights out with Thomas were what led Thomas to meet his future wife, Martha Kane. When Zatara joined the Justice Society of America he called on the team to assist the Waynes when their lives were threatened by Gotham city gangsters.
When Thomas and Martha were murdered in Crime Alley, Zatara blamed himself for not being there to save them and for Bruce becoming an orphan. This guilt, combined with the apparent death of his wife and Zatanna's mother Sindella, drove Zatara to alcoholism. When a young Bruce Wayne arrived on his doorstep, Zatara trained Bruce in the art of escapology and illusions and Bruce helped him overcome his alcoholism.
Zatara met his end in Swamp Thing #50 by Alan Moore and Steve Bissette. When Zatara insisted on taking part in a séance to protect his daughter, he took the effects of a fiery death intended for Zatanna, saving her life. Since then Zatanna has been going about her magical journey on her own, without the wisdom of her father that helped her become the hero she is today.
In addition to the history Batman shares with her father, the two were childhood friends thanks to the relationship between their parents. After the death of his parents, Bruce cut ties with Zatanna much as he did with almost everyone at the time. Their time together while Batman trained with Zatara as well as their time together with the Justice League also led them to become closer, sometimes even romantically.
However, in this story Batman explicitly states that the reason he's protective of Zatanna is because he owes Zatara, not because of his relationship with her. Given the magician's history with the Wayne family, Zatara may disagree with Bruce's assessment here. His guilt over being unable to save Thomas and Martha Wayne as well as how Bruce helped him overcome his alcoholism would suggest that he might feel he owes Bruce instead.
Still, there are a number of reasons why Bruce could feel like he owes Zatara. One would be for the training the magician gave him all those years ago. Another reason could have to do with the friendship their parents shared and the help Zatara gave them over the years. It could also be that Bruce feels guilt over how he's treated his mentor's daughter in the past, both personally and professionally. All of these things could be weighing on Batman and with Zatara gone, the debt he feels he owes, therefore, passes to Zatanna.