Boris Johnson has no plans for fresh legislation to safeguard EU citizens’ rights after Brexit, despite repeatedly highlighting the issue during the Tory leadership campaign.

In his first statement in the House of Commons since becoming prime minister, Johnson promised to ensure that the 3.2 million EU citizens in the UK can remain after Britain leaves.

He said he wanted to “repeat unequivocally our guarantee to the 3.2 million EU nationals now living and working among us. I thank them for their contribution to our society – and for their patience – and I can assure them that under this government they will have the absolute certainty of the right to live and remain.”

But afterward, the prime minister’s official spokesman suggested he had not meant to imply that there would be new legislation, or any change to the existing policy.

“You heard him talk about the settlement scheme there. We want people to be registered, and we’re making an unequivocal commitment to make sure their rights are protected. I don’t think there’s anything specific,” he said.

“Via the settlement scheme, rights for EU citizens are already protected. He’s giving an unequivocal commitment. It’s important to send the message that the 3.1 million people know how valued they are by the PM.”

Asked about whether any new legislation would be passed, he said: “It is already in law … this already has a legal basis … We have a settlement scheme under way, that is running well.”

Campaigners for EU nationals welcomed Johnson’s “warm words”, but said they were disappointed he did not commit to legislation, which he promised during his leadership campaign.

“Another damp squib on citizens rights by another prime minister,” said Nicolas Hatton, co-founder of the campaign group the3million.

Johnson’s failure to commit to new laws looks like the first pledge of his leadership campaign to be broken.

In a TV interview in June he had declared: “One thing that would be right to do, and which I suggested straight after the referendum result three years ago, is take the provisions on citizenship, the offer that we make to the 3.2million EU citizens in our country and do it in a supererogatory way, do it of our own accord, pass it through parliament.”

Hatton also raised concerns about the fate of British citizens living elsewhere in the EU. “No commitment to fully protect our rights for our lifetime but vague guarantee as if it was 2016. Meanwhile, EU citizens in the UK and Brits in Europe are still the bargaining chips of the Brexit negotiations.”

The home affairs select committee recommended primary legislation to protect EU citizens’ lifetime rights, warning of a second Windrush scandal in the making with the current settled status scheme, provided for by secondary legislation that can easily be undone by a future government.

The3million says the settled status scheme could be turned into a “declaratory process” allowing EU nationals to register rather than having to apply to remain.

Alberto Costa, the Conservative MP who has campaigned for stronger protections on citizens’ rights and resigned from Theresa May’s government over the issue, has said he believes legislation is necessary.

The row cast a shadow over Johnson’s efforts to strike a different tone from his predecessor on migration. In his statement to parliament, he confirmed a campaign pledge to commission the migration advisory committee to examine the case for an “Australian-style points-based system”.

Such an approach, which Britain already applies to some categories of non-EU migration, would see decisions made on the basis of an applicant’s likely contribution to the economy, among other criteria.

“No one believes more strongly than me in the benefits of migration to our country. But I am clear that our immigration system must change. For years, politicians have promised the public an Australian-style points-based system.

“And today I will actually deliver on those promises – I will ask the migration advisory committee to conduct a review of that system as the first step in a radical rewriting of our immigration system. I am convinced that we can produce a system that the British public can have confidence in.”

Downing Street sources subsequently confirmed that meant that hitting a numerical migration target – a policy close to May’s heart – was no longer a priority. “He’s not in the numbers game,” the source said.

Johnson also suggested, in response to a question from the Labour MP Rupa Huq, that he was prepared to consider an amnesty for migrants who have been in the UK for a long period but do not have the relevant papers.

He said he had raised the idea when he was in government, but “didn’t receive an overwhelming endorsement from the previous prime minister”.

“I have to say I do think that our arrangements, theoretically being committed to the expulsion of perhaps half a million people, who do not have the correct papers and who may have been living here for many many years without being involved in any illegal activity at all: I think that legal position is anomalous, and we saw the difficulties that that occasioned.”

Referring to the “Windrush fiasco”, he said: “Yes, I do think we need to look at our arrangements for people who’ve lived and worked here for a long time, unable to enter the economy, unable to participate properly or pay taxes, without documents. We should look at it.”