WASHINGTON -- After a couple of false starts, the Senate Judiciary Committee approved a bill Thursday that would ban assault weapons, restrict the size of ammunition clips and require universal background checks on gun sales.

But in spite of passionate pleas by Sen. Dianne Feinstein, D-Calif., the bill’s sponsor, it heads to the Senate floor with no Republican support, and it may not have the backing of every Democrat. The Republican-led House of Representatives is all but certain to reject it.

“As I’ve said before, the road is uphill,” Feinstein said Thursday, after her bill cleared the panel on a party-line vote of 10-8.

She was the lead sponsor of the original assault weapons ban Congress passed in 1994 but didn’t renew 10 years later for lack of support. The political landscape has changed since then, as has the degree of public shock over recent mass shootings, including one in December that left 20 Connecticut elementary school children dead, and another more than two years ago that gravely injured former Arizona congresswoman Gabrielle Giffords.

But even that might not be enough to restrict the use of assault weapons.

“It was a little miracle that it passed the first time,” said Robert Spitzer, the chairman of the political science department at the State University of New York at Cortland, an expert on the politics of gun control. “It has long odds now.”

Feinstein’s critics, including the National Rifle Association, say that such laws do little to deter crime and infringe on the liberties of gun owners. But Feinstein, who once trained to use a gun to protect herself, said she had seen too many mass killings, starting when she became mayor of San Francisco after two of her colleagues were slain, and especially shootings that took place at universities, office towers, movie theaters and elementary schools, as well as violence directed at police officers.

“I think a lot of my passion comes from just what I’ve seen on the streets of cities in this country,” she said.

Thursday’s vote came three months to the day after 20-year-old Adam Lanza walked into the Sandy Hook Elementary School in Newtown, Conn., and killed 20 children and six adults with an assault rifle before killing himself.

Feinstein said that her bill, which bans 157 firearms, still allowed people to buy plenty of guns.

“It exempts 2,271 weapons,” Feinstein said. “Isn’t that enough for the people in the United States? Do they need a bazooka?”

But the heated exchanges on Feinstein’s bill, and the party-line vote to send it to the full Senate, illustrate the difficulties of coming to an agreement.

“I wish we could all come a little more to the middle on this issue,” said Sen. Charles Schumer, D-N.Y.

Feinstein’s intensity was on display when Sen. Ted Cruz, R-Texas, the state’s junior senator and a tea party favorite whose confrontational style has gained attention, raised constitutional questions about her bill. He and other Republicans regard it as an intrusion on the Second Amendment.

“Would she consider it constitutional for Congress to specify that the First Amendment shall apply only to the following books, and shall not apply to the books that Congress has deemed outside the protection of the Bill of rights?” he asked.

“Likewise,” he continued, “would she think that the Fourth Amendment’s protections against searches and seizures could properly apply only to the following specified individuals and not to the individuals that Congress has deemed outside the protection of the bill of rights?”