Barack Obama has vigorously defended his right to take military action in Libya without the formal approval of Congress, after Republican leaders challenged his authority amid the right's growing suspicion of costly foreign military operations.

This week the Republican leader of the House of Representatives, John Boehner, wrote to Obama telling him that, under the 1973 war powers act, the president was obliged to seek congressional approval for the Libyan venture before Friday.

The White House replied by saying the law, which says there must be a vote in the legislature within 90 days of the president taking the US to war, did not apply because US participation in the Nato bombing did not amount to full-blown war.

The issue unites liberals opposed to foreign ventures with fiscal conservatives, including some Tea Party supporters who want big cuts in military spending, although other Republicans want defence protected from budgetary restraints. The dispute is unlikely to have an immediate effect on the US role in Libya, but is becoming a sharp issue as next year's presidential election draws near, with some members of Congress filing a lawsuit accusing Obama of breaking the law.

Hillary Clinton, the secretary of state, said she was deeply concerned by reports of wide-scale rape in Libya and other acts of sexual violence across the Arab region. She praised the "brave women" of Libya who came forward to tell of brutality at the hands of Gaddafi's forces. She urged a thorough investigation and the perpetrators to be brought to justice.

Boehner, the speaker of the House, added to that pressure earlier this week when he wrote to the president to tell him that refusing to comply with a congressional request to seek authorisation for military action in Libya appeared to violate the war powers act.

"The combination of [White House] actions has left many members of Congress, as well as the American people, frustrated by the lack of clarity over the administration's strategic policies, by a refusal to acknowledge and respect the role of the Congress, and by a refusal to comply with the basic tenets of the War Powers Resolution," Boehner said in the letter.

The White House has responded with a 38-page report to members of Congress, describing the Libya operation not as war, but a mission to remove Muammar Gaddafi from power.

The administration says that since Nato took over command of the operation in April, the US role has largely been restricted to supporting military action by Britain, France and others with refuelling and surveillance missions. But it acknowledges that remote control drones, as deployed in Pakistan and Yemen, are also at times used to fire missiles.

"US operations do not involve sustained fighting or active exchanges of fire with hostile forces, nor do they involve US ground troops," the report said.

Boehner dismissed the White House position on Thursday. "It doesn't pass the straight-face test in my view that we're not in the midst of hostilities," he said. "It's been four weeks since the president has talked to the American people about this mission. It's time for the president to outline for the American people why we are there, what the mission is, and what our goals are."

The House speaker said that Republican leaders were considering their options including "the power of the purse".

A major concern for the Republicans is the cost, after the wars in Iraq and Afghanistan contributed to the US's ballooning deficit. The White House report says the assault on Libya will have cost the US $1.1bn by the time the latest phase of the Nato operation ends in September. Some more liberal members of Congress say the US has no business intervening in foreign conflicts.

On Wednesday, 10 members of Congress from both parties asked the courts to order Obama to withdraw American forces from the Libya operation. The author of the lawsuit, Dennis Kucinich, said the White House's arguments about the degree of involvement did not stand up to scrutiny. "Look, we're at war. There's already been $750m spent," he said. "Whether there are boots on the ground or not doesn't really get into the question of whether or not the president had the ability [to intervene] in the first place. It's a constitutional issue here, and it can't be danced around at all."

Another member of Congress, California Democrat Lynn Woolsey, accused the administration of showing "contempt for the constitution".

The war powers act was passed in 1973 amid a backlash against abuse of presidential authority during the Vietnam war – including the secret and illegal bombing of Cambodia – and over the veto of President Richard Nixon.

It requires the president to notify Congress within 48 hours of sending the military in to action. If those forces are to remain in action for more than 90 days, the legislation requires that the president seek the approval of Congress.

It has been ignored by several presidents, and some administrations have questioned its constitutionality. However, the Obama White House said that it recognises that the law is legal but argues that it does not apply.