The Liberal senator Eric Abetz has rejected the cross-party marriage equality bill drafted by Dean Smith, in a sign a yes vote in the marriage law survey would lead to further government infighting about which bill to use to legislate the reform.

Abetz told Guardian Australia the bill – which is co-signed by four Liberal supporters of same-sex marriage and which Labor will support to legislate marriage equality – is “not an acceptable starting point”.

He said the bill “is seriously inadequate, as parents, freedom of speech and religious freedom, along with conscientious objection, all need full protection”.

Conservatives are working on their own same-sex marriage bill, which the assistant minister to the treasurer, Michael Sukkar, has said should include a “no detriment” clause to allow people to reject same-sex marriage in all walks of life.

The marriage law survey closed on Tuesday afternoon. The Australian Bureau of Statistics will announce the results on Wednesday 15 November.

The Turnbull government has not said which bill will be used to legislate marriage equality in the event of a yes vote.

“There has not yet been an indication of which private member’s bill would be considered,” Abetz said, refusing to respond to “hypotheticals” about possible amendments to the cross-party bill.

Sukkar has said there are “multiple bills” that could be used and the former prime ministers Tony Abbott and John Howard have called for further protections for “religious freedom”.

The education minister, Simon Birmingham, said last week that the bill produced by Smith, a Liberal, out of recommendations from a Senate committee inquiry was “the most logical starting point” and warned his colleagues not to unwind discrimination law protections.

The cross-party bill allows religious ministers, celebrants and organisations to refuse to conduct weddings but stops short of the no campaign’s calls for a broad right to discriminate against same-sex weddings for private service providers.

In a speech to the anti-gay Alliance Defending Freedom last Wednesday evening in New York, Abbott warned that if same-sex marriage was not accompanied by a “wider charter of freedoms” it would have the same consequences as overseas.

He cited claims schools have had to change their teachings about marriage and family and the case of Barronelle Stutzman, a florist he said “could lose her business because of her beliefs”.

The Equality Campaign executive director, Tiernan Brady, has said the postal survey asks “if people believe in equality and want to end discrimination in our laws”.

“If the vote is yes, it is an insult and disrespectful to do the opposite,” he said. “Any attempt to unravel existing discrimination laws, which have served the Australian people well, not just LGBTI people, would be a direct repudiation of the people’s will.”

Brady said a no-detriment clause was a “nonsense” and there was no detriment when two people of the same-sex were allowed to access civil marriage.

On Tuesday the ABS released its final estimate that 78.5% of eligible voters (12.6 million people) had participated in the marriage law survey, as of last Friday.

The Guardian Essential poll found Australia appears on track for a yes vote in the survey, with 64% of people who have voted saying they have cast a yes ballot.