The former secretary of state and New York senator told thousands of students, faculty and guests at the University of Miami that Brewer's rejection of "discriminatory legislation" recognized that "inclusive leadership is really what the 21st century is all about."
The Republican bill was designed to give added protection from lawsuits to people who assert their religious beliefs in refusing service to gays.
Clinton's remarks came before an address in which she called for greater civic participation in the country, urging young people to "find ways to make sure the barriers that too often divide us are torn down once and for all.
"It is the work of this century to complete the unfinished business of making sure that every girl and boy, every woman and man, lives in societies that respect their rights no matter who they are," she said.
Clinton has been traveling the country giving speeches at various trade industry events while considering another presidential campaign. The former first lady expects to make a decision whether to run later this year.
In a question-and-answer session, Clinton urged young people to sign up for health insurance under President Barack Obama's health care law as a key deadline approaches.
"Having access to health insurance, not connected to employment, subsidized as it is under the Affordable Care Act, liberates you to choose what you want to do in your life," she said.
The White House has set an unofficial goal of 7 million enrollees by the end of March. Nearly 3.3 million people, or less than half the total, had enrolled through the end of January.
Unified in their opposition to the law, Republicans have been relentless in focusing on its problems, from complaints of canceled policies to higher insurance premiums and Obama's unilateral decision to delay for two years the requirement that small businesses cover employees.
Clinton urged Americans to "get the facts," and cited a popular provision in the law that allows young people to stay on their parents' plans until they turn 26.
"We can still disagree—and we will—but the disagreements will be based on something resembling evidence and it won't be living in an evidence-free zone where we just argue past each other all the time," she said.
Clinton also weighed in on foreign policy.
She said Venezuela "is not being well governed" and is "going backward" under President Nicolas Maduro, but stopped short of endorsing specific actions in response to the countrywide protests that the government says have left at least 15 people dead and 150 injured. She criticized the Maduro government.
"A democracy doesn't just mean an election," she said. "A democracy means a free press, protecting the rights of opponents, protecting a free economy, having an independent judiciary."
On Syria, she said the United States should stay focused on removing chemical weapons from the war-torn country but that "a lot more international pressure is going to have to be brought to finish the job."
The mission to eliminate Syria's chemical stockpile has said the Assad government has missed at least two deadlines in the past two months to remove chemicals. The United States has accused Damascus of using stalling tactics.
Clinton said she hoped the United States can continue supporting "those forces inside Syria who are not allied with the extremists and are not looking to establish a theocratic mini-state but really want to negotiate with (President Barshar) Assad to bring about a peaceful transition."
Clinton dodged a question about her own future when University of Miami president Donna Shalala, who served in President Bill Clinton's Cabinet, asked the former first lady what the "TBD"—shorthand for "to be determined"—meant in her Twitter profile.
Referring to the social media tool's limited space, Clinton said, "Well, I'd really like to but I have no characters left."
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