House Republicans on Friday urged members to seek more information about the Libyan operations instead of voting to withdraw troops.
The House is preparing to vote on two competing resolutions on the Libya mission.
Republicans on the House floor argued for Boehner's resolution, saying that while Obama failed to seek needed congressional authority for military operations in Libya, members should not vote to withdraw troops from Libya because doing so could harm NATO allies.
Boehner bluntly rejected Kucinich's amendment on the House floor early in the debate, saying it would require a "precipitous withdrawal" from Libya that could have "serious consequences" for U.S. allies.
"In my view, the gentleman's resolution goes too far," Boehner said of Kucinich's proposal. "We may have differences regarding how we got here, but we cannot turn backs on our troops and our NATO partners who have stuck by us over the last 10 years."
House Foreign Affairs Committee Chairwoman Ileana Ros-Lehtinen (R-Fla.) added that Boehner's resolution is preferred because it puts the Obama administration on notice that Congress must play a role in military decisions under the War Powers Resolution.
She also said that anger over the administration's failure to seek congressional approval for operations in Libya should not lead members to vote for a troop withdrawal.
"Members on both sides of the aisle are increasingly frustrated," she said. "Many more are outright angry about the disregard with which the president and his administration have treated Congress."
However, she added in reference to the Kucinich resolution: "We must not let our frustration with the president's contempt for Congress cloud our judgment and result in our taking action that would harm our standing, our credibility and our interests in the region."
Ros-Lehtinen said Boehner's resolution would help force the administration to answer various questions about the Libyan operation, such as questions about its scope, duration and expected costs. "It is an open question as to whether the administration simply won't tell us or whether they just don't know the answers," she said.
She also added that the House could take more steps if these answers are not satisfactory. "If this clear warning doesn't get the attention of the White House, then more forceful action may be inevitable."
Several Democrats had sharp criticism for the Boehner resolution, saying it dodges the real choice Congress should be making: whether to authorize continued operations in Libya, or whether to approve a resolution that withdraws troops.
"If the members of the House choose to pass the Speaker's 'one-chamber' resolution, it should add one finding, that we declare ourselves to be one big, constitutionally created potted plant," House Foreign Affairs Committee ranking member Howard Berman (D-Calif.) said.
"There are two choices here," Berman argued. "If the majority thinks that the president's initial efforts to stop a humanitarian catastrophe were wrong, or that current operations in Libya do not have a compelling national security rationale, it should support Mr. Kucinich's approach.
"If the majority has concerns with Mr. Kucinich's approach, as many of us do, and believes terminating military action would have grave consequences for U.S. national security, it should simply authorize the use of force in Libya incorporating the restrictions on ground forces."