WASHINGTON — Facing vocal opposition from religious leaders and an escalating political fight, the White House sought on Tuesday to ease mounting objections to a new administration rule that would require health insurance plans — including those offered by Catholic universities and charities — to offer birth control to women free of charge.

As the Republican presidential candidates and conservative leaders sought to frame the rule as showing President Obama’s insensitivity to religious beliefs, Mr. Obama’s aides promised to explore ways to make it more palatable to religious-affiliated institutions, perhaps by allowing some employers to make side insurance plans available that are not directly paid for by the institutions.

But White House officials insisted the president would not back down from his decision last month that employees at institutions affiliated with religious organizations receive access to contraceptives.

During a campaign season that was supposed to be consumed by economic concerns, the growing uproar surrounding the contraception rule and other developments — including a federal appeals court’s decision Tuesday to strike down a California ban on same-sex marriage and the protests last week that prompted an anti-cancer group to reverse a ban on grants to Planned Parenthood — showed that social issues still resonate strongly on the political stage.