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A federal appeals court on Thursday refused to shield Hobby Lobby Stores from the Obama administration's contraception mandate -- and the fines that come with it for not complying -- in a blow to the largest employer to challenge the ObamaCare rule.

In response, the Christian-owned company vowed to appeal the case to the Supreme Court.

CEO David Green, who had taken his case to the appeals court after losing in a lower-court ruling, had argued that his family would have to either "violate their faith by covering abortion-causing drugs or be exposed to severe penalties."

The mandate requires businesses and organizations, with some exceptions, to provide access to contraception coverage -- Hobby Lobby was most concerned about coverage for the morning-after pill, which some consider tantamount to an abortion-causing drug. Hobby Lobby has refused to comply, while saying the fines could add up to $1.3 million a day.

"The Green family is disappointed with this ruling," Kyle Duncan, general counsel for the Becket Fund, the group representing the family, said in a statement after Thursday's ruling. "They simply asked for a temporary halt to the mandate while their appeal goes forward, and now they must seek relief from the United States Supreme Court. The Greens will continue to make their case on appeal that this unconstitutional mandate infringes their right to earn a living while remaining true to their faith."

If the Supreme Court takes up the case, it would only be deciding narrowly on whether to give Hobby Lobby a temporary reprieve, as opposed to ruling on the merits of the mandate itself.

There are currently more than 40 cases pending against that rule, though the Supreme Court has not yet stepped into the fray.

In its ruling, the 10th Circuit Court of Appeals said the company did not prove the rule would "substantially burden" its religious freedom. Though the mandate has exemptions for religious entities like churches, the lower court ruled that Hobby Lobby is not a religious group.

The company, founded in 1972, has more than 13,000 full-time employees across more than 500 stores.

The ruling comes after another federal appeals court earlier this week reinstated a challenge to the mandate by two religious-affiliated colleges -- Wheaton College and Belmont Abbey College.

The court ruling did not overturn the contraception mandate. But it effectively put the court case on hold while requiring the Obama administration to follow through on its pledge to revise the mandate as it pertains to religious-affiliated groups.

The contraception rule does include an exemption for religious organizations -- but that exemption does not cover many religious-affiliated organizations like schools and charities.

Complaints about the narrowly tailored exemption prompted a stand-off between the Obama administration and religious groups earlier this year. As a compromise, Obama and Health and Human Services Secretary Kathleen Sebelius decided insurers -- and not the religious-affiliated organizations themselves -- would be required to offer contraceptive coverage directly, supposedly without cost-sharing.

But the Becket Fund pointed out that the administration "has not yet taken the steps necessary to make that promise legally binding." The court decision Tuesday on the colleges would require the Department of Health and Human Services to follow through.