The Supreme Court appeared divided during arguments on Monday over the constitutionality of a provision of the Affordable Care act that can require insurance companies to offer some types of preventive care for free.

At issue is a part of the 2010 health care law that established a task force that determines certain kinds of preventative health measures that insurance companies are required to cover.

Two small Christian businesses that provide health insurance to their employees, along with some Texas residents, had sued the federal government, challenging the constitutionality of the task force.

In particular, they had objected on religious grounds to the task force’s approval of no-cost H.I.V. medications, claiming the drugs “encourage and facilitate homosexual behavior.”

But the case, Kennedy v. Braidwood Management, could have broader implications for tens of millions of Americans who receive a wide array of free health care services, including cancer and diabetes screenings, medications to reduce heart disease and strokes, and eye ointment for newborns to prevent infections causing blindness.

A ruling in favor of the challengers could mean that insurers would no longer be required to offer these health services for free.