Parliament is facing a split over how to legislate same-sex marriage, with the scene set for a Senate showdown over religious protections and anti-discrimination laws on Thursday.

After Australians voted overwhelmingly to back the social reform on Wednesday, Liberal senator Dean Smith introduced his same-sex marriage bill into the Senate, co-signed by other Liberal, Labor, Greens and crossbench politicians.

The government hopes the bill – which allows religious ministers and organisations to refuse to take part in a gay wedding – will usher in same-sex marriage in Australia before Christmas.

But prominent conservatives – from John Howard to Nationals senator Matt Canavan – said on Wednesday the bill was inadequate, arguing additional religious, speech and parental choice protections were needed.

Malcolm Turnbull hopes Parliament will deal with same-sex marriage before Christmas. Photo: AAP

Former prime minister Tony Abbott, a prominent voice of the ‘No’ side, said he would not “suddenly try to frustrate” the process, adding that same-sex marriage should be legislated as quickly as possible.

“I am not going to frustrate the will of the public,” he told 2GB Radio.

“There will be no filibustering. No clever parliamentary tactics. Just an attempt to make this the best possible bill so that no one is discriminated against.”

With the Senate to begin debating Senator Smith’s bill on Thursday, Attorney-General George Brandis flagged that he would move amendments to allow civil marriage celebrants to refuse to perform gay wedding ceremonies and to ensure it is lawful for people to hold and express their own view on marriage.

Senator Brandis stressed the amendments were not the government’s position because there “is no government position”.

“Different senators will have their own views about whether the protections in Senator Smith’s bill go far enough. I myself would prefer to see them go further,” said Senator Brandis, who supported the ‘Yes’ side.

Liberal senator James Paterson withdrew his own alternative bill on Wednesday. It would have allowed any business – such a cake shop or florist – to refuse to provide services for a gay wedding.

Penny Wong broke down in tears when the result was announced on Wednesday. Photo: AAP

Speaking on Triple J radio on Wednesday afternoon, Prime Minister Malcolm Turnbull would not “guarantee” the law would be changed by Christmas.

“It is my goal and I believe it’s the same commitment from all parties, that we should get this done before Christmas. So that’s the goal,” he said.

Mr Turnbull also rejected out of hand a proposal from Liberal MP Kevin Andrews, who argued a baker of one faith should be allowed to refuse services to people of another faith having a wedding.

Mr Andrews made the remark while arguing for religious freedoms for bakers to deny service for gay weddings, but the Prime Minister labelled it “an extraordinary proposition”.

“I absolutely do not agree with Kevin Andrews that bakers should be able to discriminate against people whether it’s on the basis of their religion,” he said.

Mr Turnbull said he expected there to be some amendment to the Smith bill.

“But everyone will have the opportunity to have their say and if they want to make a change here or improve it, or correct a defect from their point of view, they’ve got the opportunity to do it.”

Labor has indicated a desire to back Senator Smith’s bill without amendments, saying it would not back any legislation that created new forms of discrimination.

“We have the view that the Australian people voted to lessen discrimination and not to extend it,” Senator Penny Wong told Senate.

“We also take the view that the bill does not change or take away any existing protections for religious freedoms.”