Republicans claim the Democratic stance Clinton is adopting is hypocritical, because more than a dozen states had already passed similar laws and all are similar to a religious freedom protection law that Clinton's husband Bill Clinton signed as president.
Clinton last week called it "sad" that Indiana would approve the law, which like the 1993 version is called the Religious Freedom Restoration Act.
On Wednesday, Clinton had this to say about a similar law approved by the Arkansas legislature:
The original versions of both the Indiana and Arkansas laws did not specifically refer to gay people, but the measures were widely interpreted -- including by businesses often friendly to Republicans -- as a legal shield for businesses opposed to same-sex marriage or that otherwise that refuse to serve gays and lesbians.
Indiana Gov. Mike Pence (R), who signed the state religious freedom law now under broad attack, has pointed to the 1993 federal law as the basis for his state's law. Pence has noted that Illinois approved a similar law that won the signature of Barack Obama when he was a state senator.
Democrats counter that the laws are similar only in name to the one Bill Clinton signed.
The original law was intended to protect an individual's religious expression, such as Indian peyote ceremonies or the wearing of a Muslim headscarf, but the Indiana and Arkansas law are meant to shield employers or service providers opposed to gay rights, Democrats including Sen. Charles Schumer (D-N.Y.) said this week.
White House press secretary Josh Earnest also said Pence's comparison was false, mostly because of the spirit and intent of the laws 22 years apart.
But for Hillary Clinton, the fact that the text of the laws is so similar could be uncomfortable as she begins a presidential race that will cast her as a policymaker in her own right, proud of her husband's accomplishments but independent of his legacy.
"There’s no way to say credibly that the Indiana or Arkansas law is substantially the same as the nearly 25 year-old RFRA law, and those who say so are falling for the far right’s spin," said Adrienne Watson, communications director for the pro-Clinton research group Correct The Record.
The notion that Hillary Clinton is running away from the law her husband signed is part of that spin, Watson charged.
"The Indiana and Arkansas laws are bad legislation and Hillary Clinton has battled such discrimination against the LGBT community and other minorities throughout her life," Watson said.
"They're different laws in different times with a different intent," said Clinton spokesman Nick Merrill.
Bill Clinton spokesman Matt McKenna declined comment on similarities or differences among the 1993 law and the newer legislation.
Hillary Clinton has not addressed the question head-on of whether RFRA is being misrepresented in these new statutes. But she criticized the Supreme Court's decision last year in a case involving a family-run company, Hobby Lobby, that successfully challenged the provision of contraception coverage to employees. That decision is among several in which courts have stretched the original RFRA law to apply in new situations.
"The reason that was passed and Bill signed it in the '90s was because at that point there were legitimate cases of discrimination against religions," Clinton said during an appearance last summer at the Aspen Institute.
Asked whether the high court decision was a "perverse" reading of RFRA, Clinton replied, "This is certainly a use that no one foresaw."
Times have also changed, and the context of the new laws in crucial. They were passed following the expansion of legal protections for same-sex marriage, and were seen by some backers as a way to protect businesses or service providers opposed to those unions. But they were also passed in red states at a time when mainstream public opinion has tipped in favor of same-sex unions. And at a time when businesses, including many traditionally friendly to Republican ideas, saw such legislation as a potential legal liability.
"We know these new statutes are motivated by opposition to same-sex marriage and equal opportunities for gay and lesbian people. That was not the purpose in 1993. It's clearly the purpose now," said Dale Carpenter, a law professor at the University of Minnesota Law School. "These laws are being passed in a context in which there is an organized, sophisticated and well-funded set of groups around the country challenging civil rights laws protecting gays on the grounds that some people object for religious reasons."
Pence asked the state legislature to clarify the law following opposition from business leaders, and an amendment was approved Thursday
Arkansas Gov. Asa Hutchison (R) signed a revised version of that state law Thursday. The state's most famous employer, Wal-Mart, was among the businesses that raised concerns. The new version is closer to the 1993 text, and Republican legislators said the revised legislation addresses worry that the first version provided legal cover for discrimination.
In Indiana, the House and Senate approved revisions to that state's law, the Associated Press reported.
The amended Indiana law says service providers cannot use the law as a legal defense for refusing to provide services, goods, facilities or accommodations, the AP reported. It also bars discrimination on several grounds, including sexual orientation and gender identity. The measure exempts churches and affiliated schools, along with nonprofit religious organizations.
Indiana Democratic leaders said the proposed amendment didn't go far enough and that the entire law should be repealed.