WASHINGTON — President Trump, ignoring his own troubled history with women and bragging about sexual misconduct, went after Joseph R. Biden Jr. via Twitter on Thursday, posting a video that mocked the former vice president for his handsy approach to politicking.

The 15-second clip tweeted Thursday by Mr. Trump — accompanied by the text, “WELCOME BACK JOE!” — doctored a homemade video that Mr. Biden had released the day before, in which the former vice president addressed his history of physical contact with women, some of whom have come forward in recent weeks to say his intimate behavior made them uncomfortable.

As Mr. Biden speaks to the camera, a pair of hands appears on his shoulders, and then a cartoonish image of Mr. Biden’s head pops up from behind a couch and intimately nuzzles the back of the former vice president’s head.

[Joe Biden’s hugs and kisses were well known in Washington. Now he faces a generational reckoning.]

For a week, Mr. Trump’s advisers have been watching the spectacle surrounding the accusations that Mr. Biden has touched women inappropriately with something like glee.