Antoniw, who also supports the legislation, said he expected a "totally different debate", should the matter return.
The vote would not be legally binding but would be politically difficult for Labour MPs to ignore.
The Welsh government declined to give a view on whether a vote was needed on Tuesday, but said it was in "regular contact" with the UK government to understand the bill's implications.
The Senedd in Cardiff Bay, which is also known as the Welsh Parliament, has power over how Wales' health care service is run, but not the criminal law.
Leadbeater's bill covers England and Wales - and includes provisions for doctors, the Welsh government and Wales' chief medical officer.
It would require two independent doctors - and a judge - to verify whether the person satisfies the criteria to take their own life.
Leadbeater says it will ensure the strictest safeguards anywhere in the world.
Opponents have raised concerns that people could feel pressured into ending their own lives.
Julie Morgan's motion calling for a new law to allow assisted dying was defeated in October by 26 against to 19 for - with nine abstentions.
It was not binding on the Welsh government and ministers were among those who voted against, including First Minister Eluned Morgan and Health Secretary Jeremy Miles.
Miles indicated in the debate that a further vote would likely be needed, and said the law would have "long reaching implications" for the health service.
The Cardiff North Labour MS said she expected there to be an LCM, and said the Westminster law is different to the motion debated in the Senedd which was much wider and included "incurable illness, causing unbearable suffering" which is not included in Leadbeater's proposed law.
"Obviously when the motion was rejected, it was on a slightly different motion.
"There was a lot of concern about a slippery slope. Kim Leadbeater's bill addresses that in being very specific. That might give more confidence to some people."
She said it was not clear what would happen if an LCM was rejected, and how the vote in the Commons will go.
"I think its essential there's a free vote," she said of the possible Senedd vote. "I'm sure that will happen again."
