Reading from a prepared text flashed on flat screens in the Rose Garden, Mr. Biden also appeared to reproach Mrs. Clinton for distancing herself from Mr. Obama’s record lately, as she has done on trade, Syria, Arctic drilling and other issues. “Democrats should not only defend this record and protect this record, they should run on this record,” Mr. Biden said.

Mrs. Clinton called Mr. Biden afterward to express her admiration, and former President Bill Clinton spoke with the vice president as well. “Joe Biden is a good man and a great vice president,” Mrs. Clinton said in a written statement. Praising his “passion for our country” and his “devotion to family,” she credited him for a record of fighting for the middle class. “And I’m confident that history isn’t finished with Joe Biden.”

New polls in the last few days suggested that Mrs. Clinton had regained some ground against her closest competitor, Senator Bernie Sanders of Vermont, and that Mr. Biden, if he did jump in, had lost some traction. The polls suggested his absence from the field would give her a more commanding position.

Mr. Sanders, who now emerges as the lone figure seriously challenging Mrs. Clinton for the nomination, welcomed Mr. Biden’s decision. “Joe Biden is a man who has devoted his entire life to public service and to the well being of working families and the middle class,” Mr. Sanders told reporters in New York. “He made a difficult decision based on the needs of his family and his view of his future, and I respect the decision that he made.”

Republicans, for their part, used the decision to needle Mrs. Clinton. “I think Joe Biden made correct decision for him and his family,” Donald J. Trump, the leader in most polls for the Republican presidential nomination, said on Twitter. “Personally, I would rather run against Hillary because her record is so bad.”

Mr. Biden’s ambivalence about running was rooted in raw, and understandable, emotion: By his own account, the vice president has not been entirely himself since Beau Biden died. In public appearances in recent months, he has at times appeared morose and emotionally fragile.

Through most of his vice presidency, Mr. Biden had seemed largely resigned to the idea that he would not run in 2016. But he had seen Beau, the former attorney general of Delaware, as his heir, and some close to him said Mr. Biden began thinking about running to honor his late son.