Bristol Palin's request to keep the proceedings closed was denied last week by a Superior Court judge. A temporary order had authorized the use of pseudonyms while the court was considering the request, which stated that no good "could result to the child by an onslaught of media."
Levi Johnston, the father of 1-year-old Tripp, has pushed for open court proceedings, saying he "just wants a simple case on the merits."
"I do not feel protected against Sarah Palin in a closed proceeding," Johnston said in an affidavit. "I hope that if it is open she will stay out of it. Bristol's attorney is her attorney."
Bristol Palin's petition for sole custody and child support, filed in early November, also seeks a visitation schedule for Johnston. Bristol says her ex-fiance has exercised "sporadic visitation rights."
Relations between the Palins and Johnston and his family have frequently been strained since the couple broke off their engagement after their son was born in late December 2008.
Johnston denies in court documents that he has avoided his responsibilities. He is seeking shared custody.
Bristol Palin is the eldest daughter of Sarah Palin, who resigned as Alaska governor in July.
Sarah Palin announced her daughter's pregnancy days after being named the running mate of Republican presidential candidate John McCain.
Bristol Palin's custody petition calls Johnston's recent nude photo shoot with Playgirl magazine "risque."
The document also notes that Levi's mother, Sherry Johnston, should not be allowed unsupervised visits with the baby following her drug arrest. Sherry Johnston, who is serving out most of her three-year sentence under home confinement, was sentenced last month on a guilty plea to one count of possession with intent to deliver the painkiller OxyContin.
Palin's custody petition also suggested Levi Johnston may have his own issues with substance abuse, saying he made statements about seeking "weed" on Twitter.
Johnston denies making such a statement, saying the Twitter account "is a fraud" and that he doesn't have an account on the popular online social networking site.
In a motion opposing closed proceedings, Levi Johnston's attorney, Rex Butler, argued that Bristol Palin had not shown what sort of evidence could stigmatize the child.
"This case presents a custody case with similar facts that attend open cases every day in the Alaska court system," Butler wrote.