Senator Smith says Australians have upheld their side of the bargain, with almost 80 per cent returning their survey forms, now it is up to the parliament to uphold its end.
"Australians will not tolerate delay," Senator Smith told Sky News on Sunday.
He said a bill should be brought forward in the Senate this week to kick-off the debate and allowing it to continue into the two final weeks of the parliament this year.
Only the upper house is sitting this week, while both houses will return on November 27 for two weeks.
Liberal frontbencher Christopher Pyne said it will be up to the parliament whether a bill is passed by Christmas.
"(Senator) Dean Smith's bill is as good a bill as any to start the debate," Mr Pyne, the Leader of the House, told Sky News.
Liberal backbencher Julian Leeser does not favour same-sex marriage but will respect the wishes of the Australian people.
But he says there does have to be a debate about four issues about which religious communities are concerned.
"The right to preach, the right to teach, the right to provide services and employment rights. And those are things that will have to be considered in the context of any legislation that comes forward in the next few weeks," Mr Leeser told ABC television.
Senator Smith said his private member's bill, which has already been subject to a Senate inquiry, is specifically about allowing two people to marry, and does not deal with freedom of speech, freedom of religion or parental rights.
He says religious freedom does deserve to be explored, but disagrees it should be done in the marriage bill.
The Manager of Opposition Business Tony Burke said people just want the parliament to get the change done.
"Malcolm Turnbull has dragged us through a pretty ugly debate," Mr Burke told reporters in Sydney.
"For the people who campaigned for a 'no' vote, to think they can use a 'yes' outcome as a mandate for their position is truly bizarre."
