"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, Mrs 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 Montana and North Dakota caucuses.
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.
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.