Mr Biden's comments came during a Delaware speech on Monday night after the EC officially confirmed his 2020 election victory.
Responding, Ms McEnany listed a number of the president's political grievances, including his most recent false claim that the election was "rigged".
"THREE YEAR fake Russia investigation," she wrote in an early morning tweet on Tuesday.
"BASELESS impeachment...RIGGING election with mass mail-in voting w/ no safeguards," she added, before claiming those acts were more of an "assault on democracy".
While the president himself was not directly implicated in the Russia investigation, his first national security adviser, Michael Flynn, was arrested and jailed for twice lying to the FBI about his contacts with the Putin-led country.
Mr Trump on Thanksgiving Eve announced Flynn's pardon, a move top Democrat Nancy Pelosi said was aimed at protecting those who "lie to cover up his wrongdoing".
Elsewhere, the outgoing president was in 2019 accused of pressuring Ukraine to dig up dirt on Hunter Biden, the son his presidential challenger, Joe Biden, during the presidential campaign.
He is alleged to have threatened to withold $400m of military aid to Ukraine unless it complied with his requests.
That accusation appeared to be backed up by then-White House spokesman, Mick Mulvaney, who suggested the Trump administration had quid pro quo relationship with the eastern European country.
Mr Trump was subsequently impeached by the House of Representative but later acquitted by the Senate, America's upper chamber where the Republicans have a majority.
Most recently, the president has claimed that the 2020 election was rigged, citing mass voter fraud.His legal team has yet to produce any credible evidence that this took place and has suffered multiple legal defeats challenging the results in key swing states.
A wide range of figures criticised Ms McEnany over the comments, which came as some top Republicans called on the rest of the party to "move on" from the election.
Walter Shaub, former director of the Office for Government ethics, reminded Ms McEnany that her time in the White House is drawing to a close.
He said: "Kayleigh, the Office of Government Ethics has advised that you can file a termination financial disclosure report early if you update it on your last day to disclose any new interests.
"You can also ask the ethics office about post-employment restrictions, now that you’re leaving."
Josh Marhsall, founder of Talking Points Memo, said: “Get used to it, Kayleigh. Your abuse of our great country has gone on long enough. Say Goodbye.”
Keith Boykin, a bestselling author and CNN commentator, said: "Kayleigh, I'm not surprised by this or even disappointed in you. I'm embarrassed for you."