A battle over surrogacy and abortion that drew national attention to the birth of a baby in Texas this month became a wrenching courtroom drama in Dallas on Tuesday. Both the biological mother and the surrogate told a judge they cared for the child and argued that they should be deemed the parent.

The question of parentage has become a new flashpoint in what began as a seemingly ordinary surrogacy arrangement between a California couple who had struggled to have a baby on their own and an Alaska nurse who acted as their surrogate.

That arrangement was upended after the fetus was diagnosed with a serious heart condition. Citing a clause in the contract, the couple, Nausheen Gilkar and Omar Ahmed, asked the surrogate, McKenna West, to get an abortion. Ms. West refused and decided to go to Texas and have the baby.

The hearing, which stretched more than four hours, centered on the efforts by Ms. West to gain custody over the baby, a boy who was born with a severe heart condition on Aug. 12. She sought to reverse a ruling by a California court from June that recognized Ms. Gilkar and Mr. Ahmed as the parents.

Fighting back tears, Ms. Gilkar testified that Ms. West had no right to custody of the child. “She’s unstable,” she said. “She’s making us run from state to state.”

Ms. West, in her testimony, said she deserved custody in part because it was her decisions that had ensured that the baby was born at all. “He is here and he has a birthday because of the fight I have put up,” she said.

The judge, Ashley Wysocki, ended the hearing at the Dallas County Courthouse by extending a temporary restraining order while she decides the case. The order bars Ms. West from presenting herself as a parent of the baby.

After Ms. West went to Texas, Ms. Gilkar and Mr. Ahmed eventually followed her there. By the time the baby was born in Dallas, Ms. West had a legal team supported by national anti-abortion activists from the Alliance Defending Freedom, a conservative Christian legal advocacy group. The Texas attorney general, Ken Paxton, also intervened in the case to try to compel medical treatments for the child after the birth.

On Tuesday, with the 13-day-old boy still in critical condition in a Dallas hospital, Ms. Gilkar and Ms. McKenna offered tense, emotional and at times deeply personal testimony of how their surrogacy arrangement had begun, and how it had gone awry.

“We went through eight rounds of I.V.F. — my body gave up after that, and I had to have a hysterectomy,” Ms. Gilkar said, referring to in vitro fertilization, when asked by her lawyer why she had chosen surrogacy. “It was not my first choice. It was my only option.”

She said she and her husband, who live in Los Angeles, had found Ms. West through an agency, Worldwide Surrogacy Specialists, and that at first everything had proceeded normally.

Ms. West stayed with them at their home as she met with fertility doctors in California. Ms. Gilkar said she had sat by Ms. West’s side during the procedure transferring the embryo — biologically Ms. Gilkar’s and Mr. Ahmed’s — so that Ms. West could carry it to term.

“I was holding her hand,” Ms. Gilkar said through tears on the witness stand.

The relationship changed after April 17, with the discovery at around 20 weeks of pregnancy that the fetus had hypoplastic left heart syndrome, a rare and life-threatening congenital heart condition that requires open-heart surgery soon after birth, followed by two additional childhood heart surgeries.

The surrogacy contract had a clause that dealt with abortion and left the decision in the hands of Ms. Gilkar and Mr. Ahmed.

“We requested her to terminate,” Ms. Gilkar said when asked about abortion by Jeff Domen, a lawyer for Ms. West.

Ms. West initially agreed, and an appointment was scheduled, Ms. West later testified. But she changed her mind, she said, offering to care for the baby herself or put him up for adoption.

“He deserved a chance,” she said.

When questioning Ms. Gilkar, Mr. Domen pressed on that point, appearing to argue that seeking an abortion had undercut Ms. Gilkar’s claim to custody now that the boy had been born.

Ms. West, in her testimony, said Ms. Gilkar and Mr. Ahmed had tried to persuade her to get an abortion by saying that if she did not, she could be held liable for “potentially $250,000.” She said the couple were currently seeking to hold her liable in a different case.

After she decided not to go along with the abortion, Ms. West received support from a friend who connected her with anti-abortion activists. She testified that she had been living for free in a house north of Dallas and that she had raised more than $128,000 through an online fund-raising platform.

In his closing, Lee Budner, a lawyer for Ms. Gilkar and Mr. Ahmed, accused Ms. West and her supporters of trying to “eradicate surrogacy.”

The hearing on Tuesday followed an intervention in the case by Mr. Paxton, the attorney general, who had asked for a court order to compel Ms. Gilkar and Mr. Ahmed to provide lifesaving treatments to the baby after its birth. By that point, the couple was already working with a medical team on treatments for the baby.

After the attorney general’s intervention, Judge Wysocki issued an order this month along the lines of what Mr. Paxton had requested, including appointing a guardian to represent the child’s interests before the court. The judge also issued the restraining order for Ms. West.

At several points during Tuesday’s hearing, the child’s guardian, Susan Duesler, was aligned with Mr. Budner in objecting to statements and lines of questioning by the lawyers for Ms. West.

The guardian also said she was upset by statements by Ms. West in the media, such as an opinion essay by Ms. West that was published last week in The New York Post. She asked that the case file be sealed, a request that the judge granted on Tuesday.

Ms. Duesler also offered a medical update on the boy, saying he would be in critical care for at least 30 more days. She said that it would be a “minimum” of three to four weeks before the first heart procedure could be declared a success. She did not object to his eventually being transported out of the state, if medically allowed.

The two parties in the case differed during the hearing even on what to call the baby. Ms. Gilkar and Mr. Ahmed gave him the name Rumi, which their lawyer said had been put on his birth certificate. Ms. West’s lawyers used a different name, chosen by Ms. West, until they were told by the judge to stop.