Harris, 51, said her focus is on her current job and her campaign to succeed retiring Democratic Sen. Barbara Boxer.
"I'm not putting my name in for consideration. I do not wish to be considered. I am running for the United States Senate," Harris emphatically told reporters after the union rally.
Harris didn't say who she wants the president to nominate, but suggested it should be someone with "practical life experience."
She also would favor a nominee who would protect abortion rights, and marriage equality for same-sex couples, she said.
"Maybe I'm biased, but I'd like to see someone who's actually seen the impact of the court and the rulings of the court. Someone who's thinking of it not just in a way that is theoretical, but … how these laws and these rulings affect real people," she said.
Harris used herself as an example, saying that she never would have been elected were it not for the not for the educational opportunities she received because of the 1954 Brown vs. Board of Education Supreme Court ruling that found segregation in public schools to be unconstitutional.
Harris said that ruling allowed her to be a member of the second class that integrated Berkeley public schools in the 1960s.
At the SEIU union hall, Harris also threw a jab at the current Supreme Court. She criticized the court's 2013 ruling that struck down a key part of the historic Voting Rights Act of 1965, ending federal oversight of election laws in Southern states. Harris said the court "gutted it" and she vowed, if elected to the Senate, to work to reinstate those voter protections that civil rights advocates credit for with transforming the South by ensuring blacks could vote.
The attorney general also criticized members of the Senate Republican leadership who vowed to block any Supreme Court nominee put forth by the Democratic president.
"I think the Republicans have been outrageous on this issue. Outrageous," Harris said. "This president is going to be in office through January of next year. We, as Americans, deserve to have a fully-staffed United States Supreme Court. There are very important issues before the Supreme Court right now."
California's primary is set for June 7. The top two vote-getters, regardless of party, advance to the general election.
Harris' name as a possible Supreme Court nominee arose shortly after the death of Justice Antonin Scalia on Saturday.
As a sign of how the attention was intensifying over the three-day holiday weekend, Harris addressed the speculation almost immediately when she appeared before the union members on Tuesday, insisting she "was not putting my name out there."
It was her first public event since Scalia's death.
On Tuesday afternoon, Harris again was asked about the issue after she presented the Attorney General's annual California Data Breach Report at Stanford University.
"I'm not interested," she politely told an inquiring television news reporter.
As the daughter of immigrants from India and Jamaica, Harris is an appealing prospect for those pushing for more diversity on the Supreme Court.
But it would have been extremely difficult for a liberal politician from California to survive what is expected to be a bruising confirmation process in the Republican-led Senate.
Leading in the polls and with two victories in statewide elections under her belt, Harris is the front-runner in the Senate race. Her top Democratic rival is Rep. Loretta Sanchez of Orange. Their Republican challengers include Tom Del Beccaro and George "Duf" Sundheim, both Bay Area attorneys who were former chairmen of the California Republican Party.
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