WASHINGTON (AP) - Democratic supporters of a new immigration bill accused opponents Monday of trying to "exploit" the Boston Marathon bombings to hold up the legislation, sparking a testy exchange at a Senate hearing. Sen. Chuck Grassley, R-Iowa, interjected as Sen. Chuck Schumer, D-N.Y., a lead author of the bill, criticized "those who are pointing to what happened, the terrible tragedy in Boston, as a, I would say, excuse for not doing a bill or delaying it." Schumer said he wasn't talking about Grassley, who said last week that the bombings, allegedly carried out by two immigrant brothers, raised question about gaps in the U.S. immigration system that should be examined in context of the new bill. The exchange came as the Judiciary Committee opened its second hearing on sweeping legislation to strengthen border security, allow tens of thousands of new high- and low-skilled workers into the country, require all employers to check their workers' legal status, and provide an eventual path to citizenship for some 11 million immigrants now here illegally. The obstacles to the legislation, released last week by a group of four Republican and four Democratic senators, were on stark display Monday. Polls show majority public backing for comprehensive legislation including a path to citizenship, and many Republicans also support such an approach. But in some corners, opposition has not wavered. That became clear as GOP senators took turns offering critiques. Sen. Ted Cruz, R-Texas, called a path to citizenship "divisive," and said that "any bill that insists upon that jeopardizes the likelihood of passing any immigration reform bill." Sen. Jeff Sessions, R-Ala., contended that the new bill would drive down wages and eliminate jobs for American workers. Sen. John Cornyn, R-Texas, said the border security piece of the bill "falls well short of the sponsors' aspiration to protect the borders and maintain U.S. sovereignty." And Grassley said new requirements mandating employers to verify employees' legal status are ineffective. Republicans weren't the only ones to find the legislation wanting. Several Democrats expressed concerns over the exclusion of provisions to recognize gay marriages for immigration purposes. Sen. Al Franken, D-Minn., promised to fight to get such a measure included - something Sen. John McCain, R-Ariz., has said could sink the bill. Rows of spectators looked on wearing white T-shirts reading "Keep Families Together" as senators heard Monday from business and labor leaders, immigration advocates and opponents of reform, and others. But as happened at the first hearing on the bill, on Friday, the Boston Marathon bombings colored the proceedings. The attacks were allegedly carried out by two ethnic Chechen immigrant brothers who both arrived legally in the U.S. about a decade ago. One was a legal permanent resident and the other a naturalized U.S. citizen. On Monday, Sen. Rand Paul, R-Ky., joined Grassley and others who've suggested that the bombings showed the need to examine national security and the U.S. immigration system. "We should not proceed until we understand the specific failures of our immigration system," Paul wrote in a letter to Majority Leader Harry Reid, D-Nev. Paul said that national security protections must be part of any immigration legislation to ensure the federal government does everything it can to keep immigrants "with malicious intent" from using the immigration system to enter the country to commit acts of terror. Some Democrats suggest that the true motive behind at least some voicing such concerns is to oppose immigration legislation. Leahy used part of his opening statement Monday to assert that opponents of immigration reform had begun "to exploit the Boston Marathon bombing." And Sen. Marco Rubio, R-Fla., an author of the immigration bill who has strong ties to conservatives, disputed Leahy's comments. I disagree with those who say that the terrorist attack in Boston has no bearing on the immigration debate. Any immigration reform we pursue should make our country safer and more secure," Rubio said. If there are flaws in our immigration system that were exposed by the attack in Boston, any immigration reform passed by Congress this year should address those flaws. At the White House, spokesman Jay Carney was asked about the issue and said the administration believes that "one of the reasons we need comprehensive immigration reform is because it will enhance, when implemented, our national security." House Speaker John Boehner, R-Ohio, also weighed in Monday, telling an interviewer on Fox News' "American Newsroom," "I'm in the camp of, if we fix our immigration system, it may actually help us understand who all is here, why they're here, and what legal status they have." ___ (equals) Associated Press writer Jim Kuhnhenn contributed to this report.