(CNN) Ohio lawmakers for the second time this week have passed a bill that would restrict abortions, the latest of which would ban the procedure after 20 weeks' gestation.

The state's current law generally bans abortions after a fetus has begun its 20th week of gestation , unless a doctor determines that the fetus isn't viable outside the womb. Exceptions are made if the pregnancy puts the woman's health at serious risk.

The latest measure, Senate Bill 127 , would eliminate the viability test and simply ban abortions past 20 weeks. The woman's-health exception would still apply.

Gov. John Kasich will now have before him that measure, which passed Thursday, and the "Heartbeat Bill," which passed on Tuesday. The "Heartbeat Bill" aims to ban abortions from the moment the heartbeat of a fetus can be detected -- which usually occurs about six weeks into a pregnancy.

Some Ohio legislators criticized the latest measure.

"It's not about the babies; it's about the attack on women," Democratic Rep. Teresa Fedor said.

Fedor proposed an amendment to SB 127 that would have made an exception for victims of rape and incest. But the amendment failed, and the House passed the bill Thursday with a 64-29 vote.

The American Civil Liberties Union of Ohio said SB 127 would harm Ohio women. The bill would prevent those "who are facing the difficult decision to terminate a wanted pregnancy from receiving the care they need in our state," the group said in a statement published on its website.

The law "would effectively eliminate all options for women without the financial resources to travel for abortion care," the ACLU said.

In a statement, Iris E. Harvey, president and CEO of Planned Parenthood Advocates of Ohio, said: "For the second time in a week, the Ohio Legislature has inserted itself into women's private and personal health care decisions."

"These bans are a deliberate attempt to make abortion illegal in the state of Ohio," she said. "If signed into law, these bills would force women to travel long distances and cross state lines to access abortion. For many women, the expense and time these restrictions would force upon them would make access impossible."

Passed in the lame duck period

Once the new bills reach Kasich, he will have 10 days to decide whether to sign or veto them. A veto would stop a bill unless three-fifths of the state House and Senate vote for an override.

The bills were proposed at the end of the state's House session -- known as the "lame-duck" period.

"A hallmark of lame duck is a flood of bills, including bills inside of bills," Kasich press secretary Emmalee Kalmbach told CNN, referring to a larger child abuse bill within which the "Heartbeat Bill" is contained. "We will closely examine everything we receive."

Consensus to move forward

Donald Trump's election, and a presumption that he'll appoint conservative Supreme Court justices, spurred Ohio Republicans to pass the "Heartbeat Bill," which would effectively be the nation's strictest time-based abortion law, according to a legislator.

State legislators had considered the bill in previous years but it never passed the Senate.

"One, a new President, new Supreme Court justice appointees change the dynamic, and that there was a consensus in our caucus to move forward," Ohio Senate President Keith Faber, a Republican from Celina, told reporters Tuesday.

Asked if he thought the bill would survive a legal challenge, he said: "I think it has a better chance than it did before."

Should either bill become law, a court battle likely would ensue. The ACLU of Ohio already has said it would press a legal challenge.

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How does this compare to other states' laws?

Forty-three states currently restrict abortions, with some exceptions, by time or phase of fetal development.

Some of those states prohibit abortions after a doctor determines the fetus is viable. There's no fixed time period in such laws, but the nonprofit American Pregnancy Association says that viability generally can begin as soon as 24 weeks.

Other states prohibit abortion after a certain time following conception -- as soon as 20 weeks in some states.

Because heartbeats develop at around six weeks, Ohio's "Heartbeat Bill" appears to offer what would be the country's shortest window for abortions.

At least two other states -- Arkansas and North Dakota -- passed fetal heartbeat abortion laws. But those measures were found to be unconstitutional in federal court.

Where does Kasich stand on abortion?

Kalmbach, a former Ohio Right to Life staffer, said previously the governor does not typically comment on pending legislation.

Earlier this year, Kasich told CNN that he was "pro-life with the exceptions of rape, incest and the life of the mother."

Kasich signed the state's current viability restriction in 2011.

And earlier this year, Kasich signed a bill to ban the state from contracting for health services with any organization that performs or promotes abortions -- a measure widely seen as a way to defund Planned Parenthood.

Mike Gonidakis, president of Ohio Right to Life, has said his group is neutral on the heartbeat bill because it doesn't believe the measure would hold up in federal court.

"We believe in an incremental approach," Gonidakis has said. "When you overreach, sometimes the courts get the last say. There's a reason why no state has a 'Heartbeat Bill' yet."

Gonidakis previously said he believes the 20-week bill has a greater chance of surviving court challenges, in part because other states have such a restriction.

'Expensive lawsuits'

The ACLU of Ohio tweeted this week: "Just a reminder, if the unconstitutional #HeartBeatBill passes and becomes law, we will challenge it in court."

Ohio state Sen. Joe Schiavoni, a Democrat from Boardman, said that the heartbeat bill would lead to "expensive lawsuits" that would divert resources away from more pressing issues like the opioid crisis.

"To the taxpayers of Ohio, I am sorry that your money will have to be used to defend this bill in the court system."

John Fortney, a spokesman for the Senate's Republican caucus, has said that "as far as the threat of abortion advocates suing, we don't base our decisions on protecting the lives of babies on the threat of someone threatening to file a lawsuit."

Trump and the Supreme Court

President-elect Trump has expressed interest in reversing the 1973 Roe v. Wade landmark decision that made abortion legal nationwide.

Trump will get to nominate at least one justice during his term to fill the seat vacated by Justice Antonin Scalia, who died in February. Scalia was known for dissenting opinions on abortion rights.

In an interview with "60 Minutes" last month, Trump said opposition to abortion would be a criteria for nominating justices.

"The judges will be pro-life," he said.