Republican presidential candidate Ted Cruz Rafael (Ted) Edward CruzSenate Republicans face tough decision on replacing Ginsburg Cruz: Trump should nominate a Supreme Court justice next week Renewed focus on Trump's Supreme Court list after Ginsburg's death MORE is coming under fire from Democrats and progressive groups for comments he made following terror attacks in Brussels on Tuesday.

Sen. Charles Schumer Chuck SchumerPelosi orders Capitol flags at half-staff to honor Ginsburg Ginsburg in statement before her death said she wished not to be replaced until next president is sworn in Democrats call for NRA Foundation to be prohibited from receiving donations from federal employees MORE (D-N.Y.), expected to be the next Senate Democrat leader, slammed Cruz for saying law enforcement should be allowed to "patrol and secure Muslim neighborhoods before they become radicalized."

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"That's not what we do in America. We don't take a race, religion, creed, color and say we're monitoring everybody. We look for terrorism where it is, and we pursue it relentlessly. That's what we should do; that's what I believe the administration is doing," Schumer told reporters Tuesday at a press conference.

Cruz's comments came hours after a series of attacks killed dozens in Brussels, with the Islamic State in Iraq and Syria (ISIS) quickly claiming credit.

The Texas Republican said, "Our European allies are now seeing what comes of a toxic mix of migrants who have been infiltrated by terrorists and isolated, radical Muslim neighborhoods," adding that the United States must "secure the southern border to prevent terrorist infiltration."

Civil liberty groups immediately pushed back at Cruz, saying his plan is akin to discrimination.

Credo Action Political Director Murshed Zaheed said Cruz and GOP presidential front-runner Donald Trump Donald John TrumpObama calls on Senate not to fill Ginsburg's vacancy until after election Planned Parenthood: 'The fate of our rights' depends on Ginsburg replacement Progressive group to spend M in ad campaign on Supreme Court vacancy MORE are "in a race to the bottom to gin up racism-fueled xenophobia."

"Scapegoating entire communities and encouraging Americans to be suspicious of their Muslim neighbors may score points with some Republican primary voters, but it won’t do anything to make us safer,” Zaheed added in a statement.

The group previously launched a petition calling on the Republican presidential field to "denounce Islamophobia and stop spreading head."

Nihad Awad, the executive director of the Council on American-Islamic Relations (CAIR), warned that Cruz’s plan is “unconstitutional, un-American and dangerous.”

“It’s shocking that a leading presidential candidate in our nation in 2016 would suggest monitoring a section of our American society just because of its religious faith,” Awad told The Hill on Tuesday. “He claims in his speeches to defend religious liberty, but he seems to be fake when it comes to this.”

Awad linked the comments to Cruz’s announcement last week adding Frank Gaffney — a figure who has made controversial statements about Muslims and questioned President Obama's birthplace — to his foreign policy team.

“To appoint such extremist conspiracy theorists like Frank Gaffney says a lot about his mindset and what he may implement in terms of conspiracy theories and make them into policies,” Awad said. “This is going to be devastating to our democracy."

Conservatives have been distrustful of CAIR, in part because of alleged ties to Hamas.

—Julian Hattem contributed to this report.

—This story was updated at 2:03 p.m.