Washington (CNN) -- Congressional Democrats made an impassioned plea on the steps of the U.S. Capitol on Wednesday for Arizona's new immigration law to be overturned, citing constitutional concerns and other implications for the country.
"This law must be overturned -- either legally, politically or with the economic consequences that are beginning to happen to Arizona already because of the law," said Rep. Raul Grijalva, co-chairman of the Congressional Hispanic Caucus.
The request comes amid a heated debate on Capitol Hill of a comprehensive immigration reform bill -- one of President Obama's top domestic priorities.
Grijalva, who represents Arizona in the House, said the law in his state may be particular to Arizona, but "the implications, the precedent and the Pandora's box that's being opened in Arizona applies to the rest of the nation. ... This is a harbinger of things to come."
He called on Obama and the Department of Justice to join in the fight and pursue legal options, adding that this is an issue that needs to be dealt with federally and not by individual states.
Obama told reporters Wednesday that the nation needed a comprehensive plan to deal with immigration reform, rather than what he called "shortcuts" that can polarize the nation.
"What I think is a mistake is when we start having local law enforcement officials empowered to stop people on suspicion that they may be undocumented workers," Obama said.
"It's a matter of political will," the president said. "Now, look, we've gone through a very tough year, and I've been working Congress pretty hard. So I know there may not be an appetite immediately to dive into another controversial issue. There's still work that has to be done on energy. Midterms are coming up. So I don't want us to do something just for the sake of politics that doesn't solve the problem. I want us to get together, get the best ideas on both sides, work this through, and when it's ready to go, let's move. But I think we need to start a process at least to open up a smarter, better discussion than the one that is raging right now."
Attorney General Eric Holder said Tuesday that a federal court challenge to the law isn't out of the question.
Holder said the Justice Department was working with the Department of Homeland Security to "decide exactly how we are going to react to it."
Rep. Nydia Velazquez, chairwoman of the Congressional Hispanic Caucus, said that the group of House Democrats -- including members of the Congressional Asian-American, Hispanic, Black and Progressive caucuses -- is united in opposition to the law.
"It is irresponsible for any city, state or elected official to legalize racial profiling and discrimination. That is exactly what the governor of Arizona and the Republican-controlled legislature had done," the New York Democrat said.
She added: "This shortsighted law is a step backward in our nation's ongoing struggle to provide civil rights for all. This bill will not make our borders more secure, but it will open the door to discrimination and racial profiling. [It] panders to the worst elements of our national dialogue."
Scheduled to go into effect 90 days after the close of Arizona's legislative session, the law would require immigrants to carry alien registration documents at all times and require police to question people if there's reason to suspect they're in the United States illegally. Before the law's enactment, officers could check someone's immigration status only if that person was suspected in another crime.
On Monday, Arizona Gov. Jan Brewer said an executive order accompanying the law will require additional training for officers on how to implement the law without engaging in racial profiling.
Rep. Joe Baca, a California Democrat, called for a boycott of all travel to Arizona. "I personally will not be traveling through Arizona until we repeal this legislation," he said.
Rep. Luis Gutierrez, D-Illinois, later blamed South Carolina Republican Sen. Lindsay Graham for holding up progress on efforts to pass comprehensive immigration reform this year.
Graham was upset when Senate Majority Leader Harry Reid said the Senate would take up immigration instead of climate change after completing action on financial reform.
In an interview with CNN on Wednesday, Graham said pushing an immigration bill now is fruitless because it's "not going to pass."
"To bring it up now would do a lot of damage to the future of immigration reform, something I care about, and it makes energy and climate [legislation] impossible" he said.
Graham, the only leading Republican who has been working with the White House on the contentious climate change issue, walked out of those talks Saturday.
Reid reversed course Tuesday and said the climate change bill will be considered next, meaning immigration reform was pushed back to the next spot in line.
The last major effort to overhaul immigration in 2007 fell apart in the Senate, and there are many other bills passed by the House that await Senate action.
Gutierrez predicted that immigration rallies scheduled to take place in 70 cities this weekend would generate momentum across the country to move a bill this year.
Velazquez said in the coming days or weeks, she will meet with Democrats and Republicans to develop comprehensive immigration reform legislation.
"The CHC wants a bill that secures our borders, creates a reliable employer verification system, unites families, requires immigrants to learn English and mandates that immigrants pay their fair share. The time for action is now. We cannot wait any longer."
Earlier in the day, a largely Republican group of House lawmakers said it had sent a letter to Obama asking for the National Guard to be deployed along the border between the United States and Mexico.
"We want the National Guard on the border," said Rep. Ted Poe, R-Texas. "This country protects the borders of other nations better than it protects our own border."
Poe and a handful of other lawmakers said the Mexican border has become a virtual war zone, plagued by violence from Mexican drug cartels expanding their reach into the United States. They also said border state governors have made similar requests for a National Guard deployment.
Local sheriffs and border agents are "outmanned, outgunned and outfinanced" by the cartels, Poe said.
"Right now, the cartels dominate [the border region] through violence and intimidation," said Rep. Dan Burton, R-Indiana. "It is long since time to protect our populations on the southern border."
Rep. Gabrielle Giffords, D-Arizona, said her constituents were "sick and tired" of the federal government failing to protect the border. The current situation is "completely unacceptable," she said.
Giffords defended the Arizona law. She acknowledged the concerns about the bill's constitutionality but said it is a "clear calling that the federal government needs to do a better job" securing the border.
Critics say the the new Arizona law might promote racial profiling and is in conflict with federal law.
"This is about the Constitution of the United States. And this is about making sure that people have equal protection under the law," said the Rev. Al Sharpton, a civil rights activist. "And if you are a Latino in Phoenix, you should not be subjected to having to ride around with citizenship papers any more than anyone else."
Supporters say the measure is needed because the federal government has failed to enforce its own immigration laws.
Maricopa County, Arizona, Sheriff Joe Arpaio said the new law is another tool that law enforcement has in cracking down on illegal immigration.
"And we have a big problem in this state and across our nation. Something has to be done," Arpaio told CNN's Larry King. "The federal government needs help. And we're here to help."
CNN's Evan Glass, Ed Hornick, Alan Silverleib and Deirdre Walsh contributed to this report.