The Obama administration and its opponents will this week resume the supreme court battle over President Barack Obama's signature healthcare law, the Affordable Care Act, in a case that pits the religious rights of employers against the rights of women to the birth control of their choice.

Two years after the entire law survived the justices' review by a single vote – and four years after it was passed into law – the court is hearing arguments on Tuesday in a religion-based challenge from family-owned companies that object to covering certain contraceptives in their health plans as part of the law's preventive care requirement.

Health plans must offer a range of services at no extra charge, including all forms of birth control for women that have been approved by federal regulators.

Some of the nearly 50 businesses that have sued over covering contraceptives object to paying for all forms of birth control. But the companies involved in the high court case are willing to cover most methods of contraception, as long as they can exclude drugs or devices that the government says may work after an egg has been fertilised.

The largest company among them, Hobby Lobby Stores, and the Green family that owns it, say their "religious beliefs prohibit them from providing health coverage for contraceptive drugs and devices that end human life after conception”.

The Oklahoma City-based company has more than 15,000 full-time employees in more than 600 crafts stores in 41 states. The Greens are evangelical Christians who also own Mardel, a Christian bookstore chain.

The other company is Conestoga Wood Specialties of East Earl, Pennsylvania, owned by a Mennonite family and employing 950 people in making wood cabinets.

The administration says a victory for the companies would prevent women who work for them from making decisions about birth control based on what is best for their health, not whether they can afford it. The government's supporters point to research showing that nearly one-third of women would change their contraceptive if cost were not an issue; a very effective means of birth control, the intrauterine device, can cost up to $1,000 (£600).

"Women already have an income gap. If these companies prevail, they'll have a health insurance gap, too," said Marcia Greenberger, co-president of the National Women's Law Center.

The contraceptives at issue before the court are the emergency contraceptives Plan B and ella, and two IUDs.

The government also argues that employers would be able to invoke religious objections under the 1993 Religious Freedom Restoration Act to opt out of other laws, including those governing immunisations, minimum wages and social security taxes. The supreme court previously has rejected some of these claims in cases decided before the law's enactment.

The issue is largely confined to family-controlled businesses with a small number of shareholders.

A survey by the Kaiser Family Foundation found 85% of large US employers already had offered such coverage before the healthcare law required it. There are separate lawsuits challenging the contraception provision from religiously affiliated hospitals, colleges and charities.

The federal appeals court in Denver ruled in favour of Hobby Lobby. Conestoga Wood lost its case at the federal appeals court in Philadelphia.

In many respects, Hobby Lobby is the sort of company Obama would be pointing to as he advocates for corporate responsibility and a higher minimum wage. Hobby Lobby's base pay for full-time employees is almost twice the federal minimum wage of $7.25 an hour. They are offered health insurance, dental coverage and a retirement savings plan. Hobby Lobby stores close most nights at 8pm, which the company says is aimed at allowing employees to spend more time with their families.

The Greens say they have no desire to make healthcare decisions for their employees, but neither do they want to contribute to services to which they object.

One key issue before the justices is whether profit-making corporations may assert religious beliefs under the 1993 religious freedom law or the First Amendment provision guaranteeing Americans the right to believe and worship as they choose. The court could skirt that issue by finding that the individuals who own the businesses have the right to object.

The justices still would have to decide whether the birth control requirement really impinges on religious freedom, and if so, whether the government makes a persuasive case that the policy is important and is put in place in the least objectionable way possible.

Hobby Lobby and Conestoga Wood say the burden they face is clear in the $100-a-day fine for each employee they would have to pay for not complying with the contraception provision. By contrast, businesses that choose not to offer health insurance at all can pay a tax of $2,000 a year for each employee.

One potentially underemphasised aspect of the case is that there is no requirement that employers offer health insurance. They could pay the tax, which will be cheaper in many instances, according to Georgetown University's Martin Lederman, who has advanced the argument.

But Mark Rienzi, a Catholic University professor who is on the Hobby Lobby legal team, said Hobby Lobby would be at a competitive disadvantage with other employers who offer health insurance. "Their view is and has always been that they want to take really good care of their employees and their families," Rienzi said.

The companies say they believe life begins at conception, and they oppose only birth control methods that can prevent implantation of a fertilised egg in the uterus, but not other forms of contraception. There is dispute over whether any of these contraceptives works by preventing implantation, but the administration has not raised that issue in this case.