Firebrand Liberal senator Jacinta Nampijinpa Price has blamed the ABC for her comments claiming Labor was bringing Indian migrants into Australia to boost its electoral chances, insisting she has nothing to apologise for.

The Northern Territory senator said on Wednesday sections of the Australian community were concerned at “the core number, or the type, of migrants that are coming in”.

“I think Labor like to be able to ensure that they’re going to allow those in that would ultimately support their policies, their views, and vote for them as well,” she said.

“This is Labor. Basically, it’s power at any cost. And we see that occur all over the place in terms of the way they conduct themselves.”

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Price quickly walked back the claim on Wednesday afternoon, saying it was a mistake to suggest Australia had anything other than a non-discriminatory migration policy.

But on Thursday – amid growing debate across federal parliament about immigration policies and increasing instances of far-right activists objecting to multiculturalism – Price blamed the public broadcaster for the line of questioning.

“I don’t believe I have anything to apologise about,” she said.

“It was the ABC interviewer who pushed the issue, who brought up the issue of anti-Indian migration.

“What I was doing was highlighting the fact that there is huge concern for Labor’s mass migration agenda, which is applying pressure to housing, to infrastructure, to services. Then I was further pursued on this line of talking.”

Indian Australians have been targeted by anti-immigration protesters, and in the wake of the March for Australia rallies on Sunday, federal politicians including Anne Aly and Julian Leeser expressed concern about racism directed at the community.

Price described the March for Australia rallies as “a pro-Australia march” attended by “quiet Australians” and a “demonstration of unity”.

The comments also undercut moves by the Coalition to reconnect with culturally diverse constituencies, and a bid by the shadow immigration spokesperson, Paul Scarr, to draw a line under the party’s harsh anti-immigration rhetoric deployed under Peter Dutton.

Factchecking protesters' claims of 'mass migration' into Australia – videoFactchecking protesters’ claims of ‘mass migration’ into Australia – video

The opposition leader, Sussan Ley, distanced herself from Price’s comments, calling the Indian Australian community “amazing”.

“We know how hard you work, your family values, and the contribution you make across this country. And as opposition leader, I value that incredibly,” she told Sky News.

Ley also matched comments from Anthony Albanese this week, agreeing “good people” joined the marches.

“I know many from my own electorate who marched close by,” she told the ABC.

“Unfortunately, those events were hijacked by extremists,” she said, arguing lawmakers needed to “put that to one side and look at how we should have a conversation with Australians”.

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Liberal senator Dave Sharma said Price’s comments were misguided.

“I think it was very poorly expressed and I am glad Senator Price has since clarified and withdrawn those remarks, because they were hurtful to the Indian-Australian community of which I consider myself a part.

“I don’t think we should ever be going around stereotyping any migrant community for political views or beliefs or whatever that might be. People who have decided to make a home here in Australia are Australians. That is all there is to say.”

Coalition senators including Nationals Bridget McKenzie, Matt Canavan and Ross Cadell, as well as Liberals Sarah Henderson and Alex Antic, broke with their colleagues to support a One Nation proposal for an inquiry on immigration levels in the senate on Wednesday night.

The motion was ultimately defeated 37 votes to 9.

The multicultural affairs minister, Anne Aly, told question time she had spoken with Indian community members, who said they did not feel safe or secure after the weekend rallies.

“As long as we treat people with gritted-teeth tolerance instead of mutual respect, our social cohesion will remain fragile,” she said.

“I want to say to Indian Australians, this is our message. You do not have to justify your belonging in this country. We stand with you as we always have.”

Malarndirri McCarthy, the Indigenous Australians minister, told the Senate that Aboriginal and Torres Strait Islander people were rattled by attacks on the Aboriginal flag and the storming of the Camp Sovereignty site in Melbourne by neo-Nazis.

“These thugs set out to attack migrants and First Nations people on the same day,” she said.

“They attacked newest Australians, and our oldest.”