"Out of respect for my desires as a parent and out of respect for all working mums, it is with kindness that I ask other parties involved and the media to do the same. Thank you."
It was previously reported Johansson has asked for primary custody of their three-year-old daughter Rose, but Romain is desperate to leave America and move back to his home town in France with his child following the split.
Dauriac's lawyer, Hal Mayerson, told The Associated Press that he and Dauriac were taken aback by the request because he has been the "primary parent" for Rose while Johansson has been involved with her career.
Dauriac plans to petition the court to take the child to live with him in his native France, Mayerson said, adding that Johansson will have "access to her daughter any time she wants to come to Paris".
"Mr Dauriac is tired of having to run his life and his child's life based on Ms Johansson's shooting schedule," Mayerson said.
The former couple tied the knot in 2014 and kept the details of their marital ceremony secret for many months after the special day, but it was later revealed their wedding took place on a private ranch in Montana, which saw their guests enjoy outdoors activities including horseback riding, hiking and a rodeo show.
However, the pair set tongues wagging her romance was on the rocks in 2016, and the Captain America: Civil War star - who was previously married to Ryan Reynolds - confirmed she and Romain had separated in the summer of last year.
During their break-up both of Rose's parents reportedly agreed they would co-parent her and alternate their time with their child every week, although it proved to be a complicated procedure.
Speaking previously, a source said: "She would take the child on these trips like when she shooting in New Zealand.
"The kid was bouncing back and forth. It can't work because [Johansson] travels so much."
Dauriac is a former journalist who now works as a curator of art shows in New York, Mayerson said.
This was Johansson's second marriage. She was previously married to actor Ryan Reynolds from 2008 to 2011.
- Bang! Showbiz with AP