Roberts, the court's chief justice, wrote the principal dissent.
"If you are among the many Americans—of whatever sexual orientation—who favor expanding same-sex marriage, by all means celebrate today's decision. Celebrate the achievement of a desired goal. Celebrate the opportunity for a new expression of commitment to a partner. Celebrate the availability of new benefits. But do not celebrate the Constitution. It had nothing to do with it," Roberts said.
In his dissent, Scalia said the ruling is a "threat to American democracy," adding that "Hubris is sometimes defined as o'erweening pride; and pride, we know, goeth before a fall. ... With each decision of ours that takes from the People a question properly left to them—with each decision that is unabashedly not based on law, but on the 'reasoned judgment' of a bare majority of this Court—we move one step closer to being reminded of our impotence."
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Shortly after the ruling's release, United Airlines praised the court, saying the ruling "is a long-awaited victory for all those who chose to take a stand for marriage equality."
"The business community was really way ahead of our political institutions on this for years and years, recognizing that for America to be great, we don't have people to waste and we have to let everyone participate and everybody play. And the business community really led the way and continues to in many of the states where we still see discrimination, where we see backlash and anti-gay laws," Sean Patrick Maloney, a member of the U.S. House of Representatives, said in a CNBC "Squawk on the Street" interview.
American Airlines also applauded the court for finding in favor of same-sex marriage. "This is a historic moment for our country and for many of American's employees," Doug Parker, the airline's chairman and CEO, said in a statement. "Today's decision reaffirms the commitment of companies like American that recognize equality is good for business and society as a whole."
Jacques Brand, CEO of Deutsche Bank North America, said in a statement, "We are thrilled that the Supreme Court has made this historic decision in favor of marriage equality and that our LGBT colleagues and friends now have equality in this fundamental aspect of life."
In a tweet, Apple CEO Tim Cook said, "Today marks a victory for equality, perseverance and love."
Democratic presidential hopeful Hillary Clinton also praised the ruling.
The ACLU also commended the top court. "Gay and lesbian couples and our families may be at peace knowing that our simple request to be treated like everyone else—that is, to be able to participate in the dignity of marriage—has finally been granted," said James Esseks, director of the ACLU's LBGT and HIV Project, in a statement.
However, Jeb Bush, a Republican presidential hopeful, said in a statement he believes the court should have allowed states to make the decision on their own.
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Reince Priebus, the Republican National Committee's chairman, also said in a statement: "The Supreme Court failed to recognize the states' constitutional role in setting marriage policy, instead finding a federal role where there is none. In doing so, they have taken power away from the states and from the people to settle the relevant issues for themselves."
"Even though the Supreme Court has spoken with finality, there remains a diversity of opinions about marriage policy—from those celebrating today's ruling to those concerned about the constitutional balance of power," he added.
Wisconsin Gov. Scott Walker, a Republican, also spoke out against the ruling, stating it is a "grave mistake."
"Five unelected judges have taken it upon themselves to redefine the institution of marriage, an institution that the author of this decision acknowledges 'has been with us for millennia,'" he said, in a statement.
Republican Sen. Marco Rubio of Florida also said in another statement that marriage should be "between one man and one woman."
"People who disagree with the traditional definition of marriage have the right to change their state laws. That is the right of our people, not the right of the unelected judges or justices of the Supreme Court. This decision short-circuits the political process that has been underway on the state level for years," Rubio added.
Mississippi Attorney General Jim Hood said that same-sex marriages could not immediately begin in the state, according to the Associated Press.
The Supreme Court said in January it would be ruling on the matter and that it would decide two questions: Whether states must allow same-sex couples to marry and whether states must recognize same-sex marriages that take place out of state.
The issue had deeply divided the court's justices, and Kennedy was the one swing vote on the matter.
According to The New York Times, Kennedy said he was concerned about changing the country's conception of marriage after so many years. However, he also said he was concerned about excluding gay families from the institution of marriage.
—CNBC's Tom Anderson and Reuters contributed to this report.