The teenager, identified in court papers as Jane Doe, is 15 weeks pregnant. Texas bars most abortions after 20 weeks.
Lawyers for the teenager said in court Friday morning that it would be difficult to find a government-approved sponsor to take custody of their client, a Central American immigrant being held in a special detention facility in Texas for minors caught entering the United States illegally.
If the government does not find a sponsor, such as an adult relative in the United States who can care for the girl, the case would revert to a lower court judge who ruled Wednesday that the government should facilitate an abortion for the teenager "without delay."
The government's appeal of that ruling led to Friday's decision. Any subsequent order by the lower court judge would also be subject to appeal.
"She's already suffered weeks of delays, which the government has no business doing." said Jennifer Dalven, a lawyer with the American Civil Liberties Union, which represents the girl.
The teen has been seeking an abortion since late September, shortly after she was apprehended and learned she was pregnant.
The appellate ruling by judges Karen LeCraft Henderson and Brett M. Kavanaugh, both nominated by Republican presidents, allows the judges to avoid issuing a hasty decision in a case that involves complex areas of immigration and abortion law.
Judge Patricia A. Millett issued a sharply worded dissent to the decision, calling the majority's ruling "wrong" and "unconstitutional."
"There are no winners in cases like these. But there sure are losers," Millett wrote. "Forcing her to continue an unwanted pregnancy just in the hopes of finding a sponsor that has not been found in the past six weeks sacrifices J.D.'s constitutional liberty, autonomy, and personal dignity for no justifiable governmental reason."
The panel's decision noted that government lawyers acknowledged that the girl, who is in the United States illegally, "possesses a constitutional right to obtain an abortion in the United States."
ACLU attorney Brigitte Amiri urged the court not to set aside its obligation to protect the teen's constitutional right just because she may eventually obtain a sponsor, and said the government is not acting in the teen's best interest.
"They are supplanting her decision about what she should do with her pregnancy," Amiri said.
The government says it has a policy of "refusing to facilitate" abortions for undocumented minors, a departure from the Obama administration, which allowed them.
The Department of Health and Human Services, which oversees undocumented minors taken into custody near the border, is trying to "promote child birth and fetal life," according to court filings in the case.
After the ruling, the agency's Administration for Children and Families said in a statement: "For however much time we are given, the Office of Refugee Resettlement and HHS will protect the well being of this minor and all children and their babies in our facilities, and we will defend human dignity for all in our care."
Officials have said the 17-year-old could voluntarily return to her home country and seek an abortion or find a sponsor in the United States.
However, during oral arguments on Friday, the government acknowledged for the first time that abortion is illegal in the girl's homeland.
Her native country has not been released by the court to protect her privacy.
"We're not putting an obstacle in her path," Catherine H. Dorsey, the government lawyer representing HHS in the case, told the judges Friday. "We're declining to facilitate an abortion."
Dorsey told the appeals court that the government had over the past month identified two potential sponsors — both relatives — but they had fallen through. The government performs background checks on potential sponsors, a process that could take weeks or months.
During oral arguments, the judges questioned the government's stance, noting that undocumented immigrants in other types of federal custody — including adults in immigration detention and federal prisons — may seek elective abortions at their own expense.
"Even if she has that right, we don't have to facilitate it," Dorsey said.
Millett also pointed out that forcing the teenager to return home might clash with her legal right to seek asylum in the United States. The teen has said that her parents abused her in her native country.
Underscoring the significance and interest in the case, Chief Judge Merrick Garland agreed on Friday to live-stream audio of the oral argument for the first time in 16 years.
About 40 people gathered Friday morning in front of the Department of Health and Human Services to demand "justice for Jane."
"The constitutional right to abortion does not depend on your immigration status," said Georgeanne Usova, legislative counsel for the ACLU.
Usova said the group wasn't just fighting for "this young woman, but every woman in government custody."
Rachel Siegel contributed to this report.