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US election 2016: Clinton hails 'milestone for women'
Six states have been voting in primaries on Tuesday but the race in California will count the most.
Mr Sanders had been hoping for a win in that state but early results indicated a significant lead for Mrs Clinton.
As the votes were counted, he told supporters he would remain in the race and contest the remaining primaries, despite acknowledging it was a "very, very steep fight".
Mr Sanders aims to sway super delegates to support him instead of Mrs Clinton at the party's convention in July, but commentators say the Vermont Senator is unlikely to succeed in his bid for the nomination.
The AP news agency reported on Monday that Mrs Clinton already had enough delegates to qualify as the Democratic nominee.
President Barack Obama called both Mrs Clinton and Mr Sanders on Tuesday, according to the White House.
He congratulated Mrs Clinton on "securing the delegates necessary to clinch the Democratic nomination for president".
Mr Sanders will visit Mr Obama at the White House on Thursday, per Mr Sanders' request, according to the White House memo. They will discuss "the significant issues at stake in this election that matter most to America's working families".
Claiming the nomination in a speech in Brooklyn, New York, Mrs Clinton said Republican presumptive nominee Donald Trump was "temperamentally unfit" to be president.
"My mother… taught me to never back down to a bully. Which turned out to be pretty good advice," she said.
Speaking to supporters in Santa Monica, California, Mr Sanders said his campaign would not support Mr Trump, "a candidate whose major theme is bigotry".
Mr Trump had earlier called on Sanders' supporters to join him after winning his party's vote in New Jersey, South Dakota, New Mexico, California and Montana.
Analysis - Anthony Zurcher, BBC News North America Reporter
In what amounted to a Democratic nomination contest victory speech, Hillary Clinton took some time to acknowledge the historic nature of her achievement. She made reference to the metaphorical glass ceiling that she has now shattered. She referenced the long struggles of the women's rights movement. And she tipped her hat to her Democratic rival, Bernie Sanders.
Then it was on to the work at hand - wrapping Donald Trump's recent controversies around his neck and pitching him into the Hudson River.
If Mrs Clinton has run a joyless primary campaign, it has been in part because she's spent much of it nurturing her built-in advantages within the Democratic Party and playing not to lose. Last week, in a foreign policy speech in San Diego, she went on the attack. And Tuesday night, she continued the broadsides. It's a role that allows her to show considerably more energy and passion.
Earlier in the evening, Mr Trump focused almost exclusively on economic issues. Gone were references to Muslim immigration bans or border walls. Instead he made an explicit pitch to Bernie Sanders supporters and other Americans disaffected by the current state of the US political system.
It was the kind of primary night speech that will be well received by Republicans politicians who have spent the last week in a cave or a coma. The rest of the party faithful will likely be more inclined to wait and see.