The Greens successfully delayed the bill until October - with the support of the opposition - in the Senate on Monday.
The Greens oppose the fund because of a lack of support for renters, and said the delay would allow Labor to negotiate rent reforms with state and territory leaders at an upcoming national cabinet meeting.
Rent freezes cannot be imposed without the support of states and territories.
Prime Minister Anthony Albanese told his partyroom colleagues the Greens had brought a "juvenile student politics approach" to negotiations that needed to be exposed.
"They say they tried to push us further, the reality is they blocked it," he said.
"Their proposal would result in less supply and less investment, which would make the problem worse."
Asked if the legislation will be brought back, Housing Minister Julie Collins said the government would be looking at "all the options available".
"What we want to do is get on with the job of delivering homes," she said in Canberra on Tuesday.
"We're working with our partners in the states and territories, local government and the community housing sector to deliver homes each and every day, and we're going to continue to do that."
A Greens partyroom meeting on Tuesday was told the decision to defer the housing bill was "unprecedented", and the government was unable to unilaterally pull the legislation from the Senate.
It was the view of the Greens that the party was not refusing to pass the legislation, instead opting to deal with it in October after the national cabinet meeting.
The Greens have called for a parliamentary inquiry into rent affordability and rights, with an interim report to be published in September, for consideration by national cabinet.
The government announced at the weekend a $2 billion commitment for social housing, to be shared among the states and territories, ahead of the housing fund bill being voted on in the upper house.
The housing fund would use investment returns to build 30,000 social and affordable homes each year for the next five years.
Independent senator David Pocock said it was disappointing the Greens and coalition had delayed the legislation, following months of negotiations where the government made concessions.
"I don't see who wins by kicking this to October," he told ABC radio.
"There is an urgency to get going on this now - housing is not going to go away as an issue.
"My view is that we now bank these wins, we get on with building new social and affordable housing our community so desperately needs across the country."
Senator Pocock said there were 300 people every day nationwide having their requests for emergency or other accommodation go unmet.
Housing advocates and the crossbench on Monday called on the Greens to support the legislation in its current form, warning Australia was facing a crisis.
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