As an African-American woman running as a progressive in traditionally red Georgia, Ms. Abrams, a former Democratic leader in the Georgia House of Representatives, was so inspiring to Democrats outside her own state that she contemplated seeking an even higher office: the presidency.

Ms. Abrams did not reveal any plans with respect to the White House in the video.

“While I still don’t know exactly what’s next for me, here’s what I do know: Democracy in America is under attack,” she said. “Voter suppression is rampant and it is real.”

She delivered the Democratic response to Mr. Trump’s State of the Union address last winter, offering herself as a spokeswoman for a pragmatic version of progressivism focused on matters of economic opportunity and health care.

Ms. Abrams’s decision may leave national Democrats scrambling to find an alternative opponent for Mr. Perdue, a staunch conservative. At least one other significant Democrat, Teresa Tomlinson, the former mayor of Columbus, Ga., has been preparing a run against Mr. Perdue, and she may now be best positioned to become her party’s leading option.

But Ms. Abrams was the most successful Democratic candidate for high office in Georgia in more than a decade, and any other contender will have to build largely from scratch the financial and political network that Ms. Abrams will now attempt to hold in reserve until she makes a decision.

In addition to contemplating campaigns for the Senate and the White House, Ms. Abrams was discussed as a possible nominee for the vice presidency. She met in March with Joseph R. Biden Jr., a front-runner for their party’s presidential nomination, and Mr. Biden’s advisers floated the idea of inviting her to run as a partner with him in the Democratic primaries.

Advisers to both Ms. Abrams and Mr. Biden denied they ever discussed such a formal deal.

Ms. Abrams’s announcement spurred a back-and-forth between the Republican and Democratic committees charged with electing their members to the Senate.