The Greens have failed in a bid to force an urgent parliamentary debate on Australia’s potential military involvement in Iraq, with both major parties arguing deployments should remain a matter for the executive government to decide.

Tony Abbott announced on Sunday that the Royal Australian Air Force would help deliver arms to Kurdish Peshmerga fighters in a bid to counter the advance of Islamic State (Isis) militants, and would also continue with food and water air drops.

The prime minister said Australia was yet to receive a formal request from the US for a greater military involvement, such as the use of Super Hornets for air strikes against Isis targets, but would consider any such proposal through normal cabinet processes and with opposition consultation.

As parliament resumed on Monday, the Greens’ leader, Christine Milne, sought to bring on an immediate debate on a motion calling for parliamentary approval to be required before Australians were deployed to Iraq.

Her motion to suspend regular Senate business to allow the debate failed by 44 votes to 13, with the Greens mustering support from only three crossbenchers: the independent senator Nick Xenophon, the Liberal Democrat David Leyonhjelm and the Palmer United party (PUP) senator Jacqui Lambie. The other PUP senators and the Motoring Enthusiast party’s Ricky Muir were absent for the vote.

Milne argued it was time the Australian parliament was brought into the debate about military deployments and given the ability to approve such an action.

She warned of “mission creep” in Iraq, saying Australia’s role began with humanitarian airdrops and had shifted to arming the Kurds and the government was signalling that it was open for requests for Super Hornets to be involved in air strikes.

“Have we learnt nothing from the engagement in Iraq in 2003,” Milne asked, arguing that war left a vacuum that allowed ethnic tensions to come to the surface.

“The point here is very few people believe that the prime minister of Australia has got a strategic plan for Australian engagement in Iraq. Everybody believes we are simply running behind President [Barack] Obama who himself said last week he doesn’t have a strategy.”

But the defence minister, David Johnston, said military decisions were not made lightly, and the normal procedure was that the prime minister and cabinet made a recommendation to the governor general.

“This is the way we have always done our business,” he told the Senate. “No government takes the putting of young Australians in harm’s way other than with the utmost seriousness. This is probably the most important decision a prime minister can ever make.”

Johnston spoke of the dangers of Isis, saying it had committed mass executions and was responsible for behaviour that was “tantamount to ethnic cleansing” as it moved through Iraq “with lightning speed”.

He said the only force that had provided any strong resistance to Isis was the Kurdish forces in north-eastern Iraq. The minister said he would not like to see that resistance fail through a lack of military equipment.

Labor’s defence spokesman, Stephen Conroy, said the opposition would not support the suspension of standing orders because the Greens’ motion was a “stunt”.

Conroy accused the Greens of conflating two issues: the importance of debate and the suggestion of requiring parliamentary approval for military deployments.

He said a parliamentary vote for military action was “fraught with danger” as governments must retain maximum flexibility to respond to security threats quickly and efficiently.

“The government of the day has access to classified information which the parliament does not,” Conroy said.

A former Labor defence minister, John Faulkner, told the Senate any decision to deploy the Australian Defence Force was a great responsibility for the executive government, and he did not believe standing orders should be suspended “to demand such a debate” in the terms suggested.

“I would strongly encourage the government to be as open and as transparent as possible about this current deployment of Australian defence personnel,” Faulkner said. “I would suggest that the defence minister, Senator Johnston, make a ministerial statement on this matter as soon as he is able to do so and then I would suggest that the government facilitate full debate in this chamber around this statement.”

Leyonhjelm said the Liberal Democrats believed troop deployments should be subject to approval by a vote of two-thirds of both houses of parliament. Leaving the decision in the hands of executive government risked “adventurism” and political posturing, Leyonhjelm said.

The Greens senator Scott Ludlam said both major parties seemed to be arguing “if it ain’t broke don’t fix it” but Australia was again acting “at the behest of the US government”.

The House of Representatives, meanwhile, debated a separate motion condemning the actions of Isis in Iraq “which amounts to attempted genocide of minorities including the Assyrian, Chaldean, Mandaean and Yezidi people”.

The opposition treasury spokesman, Chris Bowen, moved the motion, which also reaffirmed “the rights of the Christian and other minorities of Iraq to live in peace and freedom and calls for all steps to be taken to ensure that all members of the affected communities can live in freedom in Iraq”.

Both major parties spoke in support of the motion.

The Greens’ deputy leader, Adam Bandt, and the independent MP Andrew Wilkie failed in separate bids to bring on urgent debate in the lower house about whether Australian forces should be deployed to Iraq.

The justice minister, Michael Keenan, said the prime minister would make a statement to the parliament regarding the Iraq developments later on Monday.