SAN FRANCISCO -- Officials have determined that San Francisco will remain an immigration-related sanctuary city, despite criticism after a woman was killed by a Mexican national who had been released from jail even though federal agents wanted him detained for deportation.

The Board of Supervisors unanimously approved a resolution Tuesday that urges the sheriff not to participate in a detainer-notification system that asks jails to let Immigration Customs and Enforcement officials know when an inmate of interest is being released.

The action sent a strong but symbolic message to critics who had lambasted San Francisco after the July 1 waterfront shooting of 32-year-old Kate Steinle.

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Earlier in the day, Senate Republicans in Congress tried but failed to push through legislation punishing sanctuary cities. CBS News' Stephanie Condon reported on Tuesday that the legislation in question, sponsored by Sen. David Vitter, R-Louisiana, would withhold federal funding from "sanctuary" states or cities that fail to comply with Department of Homeland Security-issued detainer requests for undocumented immigrants. Two GOP presidential candidates are co-sponsored the measure: Sens. Ted Cruz of Texas and Marco Rubio of Florida (who has largely disavowed his support for 2013 comprehensive immigration reform efforts).

An undated photo shows Kathryn Steinle, who was shot and killed July 1, 2015, in San Francisco. KPIX

The death of Steinle cast an uncomfortable spotlight on the city that proudly declares itself a refuge for immigrants. As outrage mounted nationally, U.S. Sen. Dianne Feinstein, a California Democrat, criticized San Francisco's sheriff, saying that suspect Juan Francisco Sanchez-Lopez should have been detained.

Sanchez-Lopez says he found the gun under a bench on the pier, and it accidently fired when he picked it up.

"All of us in this room agree that the death of Kathryn Steinle was senseless and tragic, but what many of us disagree on is the role - if any - that San Francisco's existing sanctuary and due-process-for-all" ordinances played in the event, Supervisor Malia Cohen said to cheers from the crowd that gathered for the meeting.

Supervisors said they wouldn't let hateful commentary undermine a long-standing policy that improves public safety and embraces immigrants.

Juan Francisco Lopez-Sanchez is led into the Hall of Justice for his arraignment in San Francisco, California July 7, 2015. REUTERS/Michael Macor

"I'm so proud of San Francisco," Supervisor David Campos, who co-sponsored the nonbinding resolution, said after the vote. "I'm so proud that notwithstanding the climate at the national level of scapegoating immigrants that San Francisco went against that."

Roy Beck, director of NumbersUSA, which calls for limiting immigration, said it's frightening that supervisors sided with immigrants who are in the country illegally - even violent ones - rather than public safety.

San Francisco declared itself a sanctuary city in 1989, passing an ordinance that bans city officials from enforcing immigration laws or asking about immigration status unless required by law or court order. A follow-up ordinance in 2013 allows detention only under a court order targeting violent felons.

San Francisco and other cities and counties have routinely ignored requests from ICE to keep people in custody. The jurisdictions say they can't hold arrestees beyond their scheduled release dates without probable cause.

However, more than half of the roughly 340 jurisdictions that previously declined to cooperate with ICE are now doing so in some form, as long as they don't have to keep immigrants in custody.

The shift came after outreach by federal immigration officials and several high-profile cases, including the one in San Francisco.

San Francisco Sheriff Ross Mirkarimi said on Tuesday through a spokeswoman that the unanimous vote by the board validates his policy.

Supervisors voted to table another resolution urging the sheriff to revoke a department-wide memo prohibiting communication between his staff and federal immigration authorities, saying it would send the wrong message to people living in the city illegally.