Seeking to accentuate a political vulnerability for President Obama, Republican lawmakers on Wednesday intensified pressure on the White House over a controversial new health-care rule that critics say violates religious liberty — vowing to press for legislation to repeal it unless the administration relents.

The specter of a legislative showdown over the law mandating coverage of prescription contraceptives threatened to expose divisions among Democrats, with former Virginia governor Timothy M. Kaine, running for Senate in a crucial presidential battleground, becoming the latest Obama ally to criticize the policy even as liberal lawmakers rushed to the president’s defense.

Kaine, a former Democratic National Committee chairman from 2009 to 2011, told radio station WHRV in Hampton Roads that the administration “made a bad decision” in how it crafted the policy, according to a transcript from his campaign.

Two Democratic senators, Robert P. Casey Jr. of Pennsylvania and Joe Manchin of West Virginia, have already called on the White House to change the policy, with Manchin slamming it as “un-American” in a letter to Obama.

White House officials, along with dozens of liberal religious leaders and several leading Democratic lawmakers, defended the policy Wednesday, describing it as a crucial protection for women who deserve birth-control coverage no matter where they work. The White House also publicized a support letter signed by 600 doctors and medical students. Nearly two dozen leaders of organizations including Catholics for Choice and the Central Conference of American Rabbis issued a statement saying the policy will “safeguard individual religious liberty” while helping “improve the health of women, their children, and families.”

The rule, enacted last month as part of Obama’s health-care overhaul, requires employers to provide female employees the full range of contraceptive coverage, including birth control, the “morning-after pill” and sterilization services. The measure exempts churches but covers religiously affiliated colleges and hospitals, meaning that many Catholic-run institutions would have to offer plans that church leaders say violate their teachings.

Catholic bishops have been leading the opposition to the rule by distributing letters and other materials to be shared with millions of worshipers.

The issue has exploded as a campaign theme, with GOP presidential candidate Mitt Romney and others accusing Obama of pursuing an anti-religion agenda.

With their statements Wednesday, GOP lawmakers sought to elevate the controversy beyond a potentially narrow dispute over birth control into a larger battle over government intrusion

“In imposing this requirement, the federal government has drifted dangerously beyond its constitutional boundaries, encroaching on religious freedom in a manner that affects millions of Americans and harms some of our nation’s most vital institutions,” House Speaker John A. Boehner (R-Ohio) said in a rare speech on the chamber floor. “If the president does not reverse the attack on religious freedom, then the Congress, acting on behalf of the American people and the Constitution we are sworn to uphold and defend, must.”

GOP lawmakers appear to see a political opportunity in the issue with Hispanics, many of whom are Catholics. The party views social issues as an entry point with Hispanics, who have been trending Democratic.

One of the senators drafting legislation to overturn the rule is Sen. Marco Rubio (R-Fla.), a Cuban American who appeared on television Wednesday speaking about it in English and Spanish.

“The vast majority of Americans, whether on the left or the right, will tell you that the government of the United States should not have the power to be able to go in and tell a church-based organization that they must pay for something that their faith teaches their members not to do,” Rubio said.

The new round of criticism came after White House officials tried to soothe tensions over the issue. Obama aides have been hearing criticism not only from the right but also from their allies who oppose the rule on policy grounds and consider it a miscue that could hurt the president with Catholic swing voters and other churchgoing independents.

Obama aides signaled a willingness to discuss a mutually agreeable way to put the new rule into practice following a phase-in period to end in August 2013. Officials from the U.S. Conference of Catholic Bishop, though, discounted the assurances.

White House press secretary Jay Carney would not say Wednesday whether the president would veto a bill to overturn the birth-control mandate. He reiterated promises to work with religious groups to address their concerns.

“There are ways to approach this that would ensure the rule is implemented so that women have access to these important health-care services, no matter where they work, but also that hopefully would allay some of the concerns expressed,” Carney said.

The matter came up briefly during a closed-door session between Obama and Democratic senators holding a retreat at Nationals Park. Several female lawmakers stood to thank the president for his stance, according to a person briefed on the meeting.

Staff writers David Nakamura and Ben Pershing contributed to this report.