WASHINGTON—It was “with a heavy heart” that Senator Patrick Leahy, the Vermont Democrat, said he was withdrawing his amendment that would have allowed gay Americans to bring their foreign partners into the country as part of the immigration overhaul that passed through the Senate Judiciary Committee earlier this week.

Leahy’s decision marked the latest in a series of setbacks for gay rights legislation on the national level. It came at a time when gay rights seems to be advancing; 53 percent of Americans said in a recent Gallup Poll that they support gay marriage, and 11 states and the District of Columbia have passed laws to allow it. But Tuesday’s action demonstrates how gay rights activists continue to face significant hurdles when specific measures are proposed on Capitol Hill.

The amendment would have granted citizenship to any immigrant who could prove to be in a monogamous, financially interdependent relationship with an American citizen. There are currently 28,574 same-sex couples living in U.S., including 1,215 in Massachusetts, in which one partner is a U.S. citizen and the other is not, according to a study by Craig Konnoth and Gary Gates of the Williams Institute at the University of California,Los Angeles.

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Republicans called the amendment a “poison pill” and threatened to sink more than 300 amendments introduced to the bill, which includes a pathway to citizenship for undocumented immigrants already in the United States.

Senator Lindsey Graham, a Republican member of the bipartisan “Gang of 8” who brokered the bill, said in an interview that he could not sell immigration reform to his constituents in South Carolina if it included gay rights. He also said it would “fracture the coalition” of immigration reform advocates in the faith-based community.

“Every emotional, tough issue has a certain amount of market. That would be beyond what the market would bear,” Graham said about the immigration measure for same-sex couples.

Faith-based groups that typically support immigration measures were also split by the amendment. The Reverend Clete Kiley, a Catholic, said introducing the same-sex couples amendment into the immigration bill would cause him to pull his support.

“I say this as a Catholic priest, that it’s another way to try to define marriage. And I ask, ‘Is that helpful in this particular part?’ ” said Kiley.

Sen. Chuck Schumer of New York took to his Facebook page after the decision to withdraw the amendment to express his regret, saying it was “one of the most trying days in my 30-plus years in public office…”

“I know that this provides little comfort,” Schumer wrote. “But I want you supporters of this amendment to know, I will be here and ready to work with you to advance the cause of legal equality. This is far from the last battle.”

Gay advocacy groups are pressuring Leahy to bring the amendment up as a stand-alone bill when the Senate begins floor debates on immigration next month, but advocates said the chance of passing a bill like it is slim when not included in a broader package. The Associated Press reported that President Obama asked Leahy to withhold the amendment from the broader immigration bill, but neither Leahy nor White House spokesman Jay Carney would confirm that the conversation took place.

Without being included in broader legislation, like the immigration amendment Leahy originally proposed, gay rights bills are less likely to pass. The Employment Non-Discrimination Act, which would protect employees from being discriminated against for their sexual orientation, has been repeatedly introduced since 1994 to no avail. In 2011, Senator Kirsten Gillibrand, the New York Democrat, introduced “Every Child Deserves a Family Act” to prohibit state adoption agencies receiving federal funding from discriminating against same-sex couples, but the bill has yet to progress in the House.

Despite the setbacks, gay rights activists remain hopeful that the tide will soon change in their favor. Former Congressman Jim Kolbe, who is openly gay, voted for the Defense of Marriage Act, which bans national recognition of same-sex marriage, in 1996. Kolbe had not yet publicly acknowledged his sexual orientation.

Ten years later, Kolbe sponsored an amendment identical to Leahy’s in the 2006 immigration debate, but withdrew it believing it was impractical. Yet Kolbe said he thinks the country now would accept giving citizenship to foreign partners in same-sex couples.

“The world has changed since 2006. States have now legalized the marriage of same-sex couples. The public attitude on this has changed very dramatically,” Kolbe said.

But members of Congress like Graham, who represent traditional, conservative constituencies, would disagree.

“People in South Carolina have come to accept immigration reform as a necessity. We have a different view of traditional marriage than our good friend Senator Leahy. That debate would be for another day at another time and it’s working its way through the courts,” said Graham.

Meanwhile, gay rights activists are looking to another branch of government regarding the immigration measure: If the Supreme Court declares the Defense of Marriage Act unconstitutional when they announce their decision by the end of June, gay Americans will be allowed to marry and sponsor their foreign partners.