Hillary Clinton has thanked her supporters for helping her reach a historic moment for women - the Democratic nomination for president.

"Thanks to you, we've reached a milestone," she told cheering crowds at a rally in New York.

She hailed "the first time in our nation's history that a woman will be a major party's nominee".

Earlier Clinton won the Democratic primary in New Jersey, cementing her hold on her party's nomination.

She went on to win South Dakota and New Mexico, while her rival Bernie Sanders found victory in the North Dakota caucuses.

Six states have been voting in primaries on Tuesday but the race in California will count the most.

Sanders is hoping for a win in that state, where polls show the race is close.

He aims to sway super delegates to support him instead of 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 Clinton already had enough delegates to qualify as the Democratic nominee.

President Barack Obama called both Clinton and Sanders on Tuesday, according to the White House.

He congratulated Clinton on "securing the delegates necessary to clinch the Democratic Nomination for President".

Sanders will visit Obama at the White House on Thursday, per 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, 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.

"To every little girl who dreams big: Yes, you can be anything you want—even president. Tonight is for you," Clinton tweeted following her win in New Jersey.

Meanwhile Trump won in his party's vote in New Jersey, South Dakota, New Mexico, California and Montana.

The billionaire turned his attention to the election in November in his remarks at Trump National Golf Club in Briarcliff Manor, New York.

"We're only getting started and it's gonna be beautiful," he said.