Venegas drove herself to the hospital and delivered her baby, she told local media. Surveillance video from inside the store shows her crying and distraught.
Arrona is a Mexican national illegally living in the country and has been detained pending removal proceedings, ICE Lori Haley told the Washington Post.
"ICE continues to focus its enforcement resources on individuals who pose a threat to national security, public safety and border security. ICE conducts targeted immigration enforcement in compliance with federal law and agency policy," Haley said.
Arrona's legal representative, Emilio Amaya García, told the Post he believed Arrona was not suspected of crimes within the United States, but did not say if he was suspected of crimes elsewhere. He is representing Arrona through San Bernardino Community Service Centre, a nonprofit group.
His concern was how the arrest affected Venegas. She had preeclampsia, he said, a pregnancy complication marked by high blood pressure. The couple explained the condition to the agents, he believes.
"They left her alone," he told the Post. "They put the well-being and safety of the mother and baby at risk."
Garcia said Venegas remains in the hospital.
Haley declined to say whether the arresting agents considered delaying Arrona's arrest until after the birth or could have escorted Venegas to the hospital, given her condition. She also declined to provide any regulations either allowing or prohibiting such agent-level decisions.
ICE has ramped up arrests under the Trump Administration following Executive Orders that directed the agency to pursue any undocumented alien in the country. Given budget and resource constraints, ICE had previously used what it called "prosecutorial discretion" to bypass nonviolent offenders in favour of criminals and recent migrants.
That has changed since President Donald Trump took office.
"If you're in this country illegally, and you committed a crime by entering this country, you should be uncomfortable, then-ICE director Thomas Homan told lawmakers in June 2017. "You should look over your shoulder."
The Trump Administration has also removed protections from certain classes to align with the Executive Orders. For instance, the Administration in March rescinded an Obama-era directive that generally ordered immigration officials to release pregnant women from custody.