Fresh from addressing a conservative US lobby group, Mr Abbott says parliament should respect the people's decision, whatever it is.
Although personally against same-sex marriage, he would support the outcome, he told Sky News on Saturday.
But advocates for change are against an "expensive" plebiscite, saying the recent debate about whether MPs will respect the outcome has brought "simmering tensions" to the surface.
Some conservative MPs and senators, including dumped cabinet minister Eric Abetz, have flagged their right to a conscience vote in parliament, potentially ignoring the will of voters.
"Given that mess, surely the easiest thing to do is bring it back to parliament, have a vote, get it done and move on," Australian Marriage Equality national director Rodney Croome told ABC TV.
"My advice to Malcolm Turnbull is if you pull off the band-aid now, most people would be pleased with that.
"It will be over in a month and you can move on to the election without this being a distraction."
Opposition Leader Bill Shorten echoed the sentiment, saying Mr Abbott dreamt up a taxpayer-funded plebiscite to "kick the issue of marriage equality into the long grass of the next few years".
"We should have a conscience vote in parliament," he told reporters in Brisbane, reiterating a Labor government would introduce legislation for marriage equality within its first 100 days.
But the Australian Christian Lobby, which argues against changing traditional marriage, says the $160 million price tag of the promised plebiscite is reasonable to settle "a long running campaign" by activists.
"They keep claiming overwhelming support yet don't want to take the issue to the people. This doesn't make sense," ACL managing director Lyle Shelton said in a statement.
In his address to the Alliance Defending Freedom in New York, Mr Abbott said he believed it would be easier for Australians who felt strongly about the issue to accept a decision if it were made by the people.
If the government is returned at the next election MPs who support same sex marriage will be asked to finalise a bill, which will then be put to the people at a plebiscite.
"This is the best way to decide something that's so important but so personal," he said.
"It's to let the people decide so that the decision, whichever way it goes, will have their authority."
