Fultz's ex-girlfriend has taken him to court for harassment and violation of privacy. A domestic court official has recommended the billboard be removed.
But Fultz's attorney argues the order violates his client's free speech rights.
"As distasteful and offensive as the sign may be to some, for over 200 years in this country the First Amendment protects distasteful and offensive speech," Todd Holmes said.
The woman's friends say she had a miscarriage, not an abortion, according to a report in the Albuquerque Journal.
Holmes disputes that, saying his case is based on the accuracy of his client's statement.
"My argument is: What Fultz said is the truth," Holmes said.
The woman's lawyer said she had not discussed the pregnancy with her client. But for Ellen Jessen, whether her client had a miscarriage or an abortion is not the point. The central issue is her client's privacy and the fact that the billboard has caused severe emotional distress, Jessen said.
"Her private life is not a matter of public interest," she told the Alamogordo Daily News.
Jessen says her client's ex-boyfriend has crossed the line.
"Nobody is stopping him from talking about father's rights. ... but a person can't invade someone's private life."
For his part, Holmes invoked the U.S. Supreme Court decision from earlier this year concerning the Westboro Baptist Church, which is known for its anti-gay protests at military funerals and other high-profile events. He believes the high court's decision to allow the protests, as hurtful as they are, is grounds for his client to put up the abortion billboard.
"Very unpopular offensive speech," he told the Alamogordo Daily News. "The Supreme Court, in an 8 to 1 decision, said that is protected speech."
Holmes says he is going to fight the order to remove the billboard through a District Court appeal.