Correction: A previous version of this story inaccurately said that Sen. Robert Menendez (D-N.J.) called President Obama the “deporter in chief.” Menendez called on Obama to “halt needless deportations” of undocumented immigrants but did not use the term “deporter in chief.”

President Obama participates in a town hall-style forum to encourage Latino Americans to enroll in Obamacare health insurance plans Thursday at the Newseum in Washington. (Jonathan Ernst/Reuters)

 Whenever President Obama has sought to turn up pressure on Republicans on issues important to Latinos, he has found reliable partners in Spanish-language TV networks Univision and Telemundo.

On Thursday, Obama came calling again — but this time, his partners became adversaries.

Obama was appearing at a town-hall-style event at the Newseum to encourage the Latino community to sign up for health-insurance policies under the Affordable Care Act. But the hosts, Jose Diaz-Balart of Telemundo and Enrique Acevedo of Univision, turned their sights on another issue: immigration.

“Your reputation has been tarnished among Latinos over deportations,” Acevedo said, referring to the administration’s removal of nearly 2 million immigrants who were in the country illegally. “How can you ask the Latino community to trust you?”

“I would challenge the premise,” Obama shot back testily as he sat onstage before the live audience of 150. Citing his work on immigration reform, affordable housing and health care, the president added: “I think the community would understand that I’ve got their back and I’m fighting for them.”

The sharp exchange — one of several contentious moments at the Thursday event — illustrated how far some Latinos have drifted from Obama since the heady days after his November 2012 reelection, in which 71 percent of Hispanic voters supported the president over Republican challenger Mitt Romney.

That huge margin among one of the nation’s fastest-growing voting blocs marked what many immigration advocates believed would be a seminal moment. Obama, who had focused on health care in his first term, was suddenly championing comprehensive immigration reform at a time when Republicans were pledging to find ways to expand their appeal to Latinos and other minority groups.

Yet 16 months later, immigration legislation has stalled in the Republican-controlled House, producing a rising sense of frustration in the Latino community that Obama has not done more to halt deportations of illegal immigrants. He used his executive authority in 2012 to stop deportations of young immigrants brought to the United States as children, but he has balked at expanding the order.

This week, two top White House allies on immigration — Janet Murguía, president of the National Council of La Raza, and Sen. Robert Menendez (D-N.J.), co-author of a Senate-approved immigration plan — demanded that he do more to provide relief to families whose relatives have been removed from the country. In a speech, Murguía labeled Obama the “deporter in chief.”

At the Newseum, the hosts pressed Obama several times for assurances that personal information from people who signed up for insurance on federal exchanges would not be sent to immigration enforcement offices in an effort to go after undocumented family members.

In an interview, Murguía said that while her organization holds Republicans accountable for not supporting reform legislation, “we don’t believe that the two are mutually exclusive.”

“We are simply pointing out that while [Obama] does have the community’s back on comprehensive immigration reform, he needs to have our back on the unnecessary deportations,” she said.

Asked about such criticism onstage, Obama argued, as he has before, that he is powerless under federal law to further halt deportations and placed the onus on congressional Republicans to support a legislative overhaul of border-control laws.

“I am the champion in chief of comprehensive immigration reform,” Obama insisted. “But until Congress passes new laws, I am constrained in what I’m able to do.”

While some legal analysts have said that Obama could unilaterally suspend a portion of the deportations, White House officials say such a move could backfire by lending credence to Republican claims that he could not be trusted to enforce border laws as part of immigration reforms.

After the event, Obama’s Twitter account posted a message to its 42 million followers reciting the “champion in chief” phrase and including the hashtag #ActOnReform. But the sentiment was met with derision by some influential immigrant rights groups.

Obama “announced to the world that the White House is once again on the defensive about his deportation policy. But no public relations campaign can cover up the pain of our families,” Cristina Jiménez, managing director of United We Dream, said in a statement. “The White House excuses are stale, played out and have no legal standing.”

In a conference call with reporters, Rosi Carrasco, a Chicago-based activist, said that Obama’s “rhetoric does not match reality” and added: “Reality is the exercise of discretion. The president is doing nothing.”

White House aides say they remain confident of strong support for the president among Latinos, and Republicans have done little to woo them. Since the Senate approved a bipartisan immigration bill last year, House Speaker John A. Boehner (R-Ohio) has not brought immigration legislation to the floor for a vote.

A Washington Post-ABC News poll released this week found that 59 percent of Hispanics approve of Obama’s job performance while 37 percent disapprove. That’s better than Obama’s 46 percent approval among the public at large.

Gallup polling shows bigger problems for Obama, who suffered a 19-point drop in approval among Hispanics over the past year, from 73 percent to 54 percent. That’s a far steeper drop than a 7 percent decline in his Gallup standing among the general public.

On the president’s health-care law, Latinos have generally been more positive since its passage than the general public. Obama’s pitch on Thursday was aimed at encouraging young, uninsured Latinos to enroll in a health plan, which needs a broad range of consumers to succeed.

Still, the president was clearly on the defensive at the Newseum.

“When you’re the president of the United States, someone’s always frustrated you haven’t done something or not done it fast enough,” Obama said. “I understand that. It’s part of the job.”

But, he continued, “the main point I have right now is that you don’t punish me for not signing up for health care. You’re punishing yourself and your family.”

Scott Clement contributed to this report.