The president apparently watched the speech, or was told of it, because even before it was over he noted on Twitter that Mr. Biden had called for his impeachment and claimed the Bidens had “ripped off at least two countries for millions of dollars.”

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“Joe’s Failing Campaign gave him no other choice!” wrote Mr. Trump.

There is no evidence that Hunter Biden made millions of dollars from his overseas work or that his father intervened inappropriately with Ukraine or China, the other country Mr. Trump was alluding to in his tweet. The president has also urged China to look into the Bidens.

Mr. Biden again denied that he did anything improper as vice president. He noted that his 2016 call to fire Ukraine’s prosecutor general, Viktor Shokin, was part of United States government policy under President Barack Obama. It also reflected the wishes of a group of Republican senators at the time, who sought Mr. Shokin’s ouster because he was seen as unwilling to target corruption, Mr. Biden said.

A longtime senator before he became vice president, Mr. Biden is deferential to congressional prerogatives and has resisted calling for impeachment, even as Ms. Warren and many of his other Democratic rivals have been outspoken in demanding it.

Last month, when House Democrats began moving toward an impeachment inquiry, he said they should only do so if Mr. Trump did not cooperate with their investigation. Then last week he said he would let Congress “do its job on impeachment.” And at a later event on Wednesday, in Manchester, N.H., he nodded to his longstanding discomfort with impeachment, calling it “a God-awful thing for a nation to go through.”

But as Mr. Trump and his allies wage near-daily political warfare against Mr. Biden, he has sought to mount a counteroffensive.

He began his rebuttal in a speech last week in Reno, Nev., but on Wednesday he went even further. Speaking from a teleprompter and dressed formally in suit and tie, he used his first trip back to New Hampshire since the impeachment investigation got underway to both taunt and condemn Mr. Trump in remarkably stark language.