The attack came in the form of a tweet shared by the Republican on Thursday.
“Question for the sitting Vice President of the United States: Do you have any capacity for shame?” Mr Conway wrote, adding: “Because if you still do, this is the time to show it by speaking up against this madness. If you don't, your silence will never be forgotten.”
While Mr Pence has not gone as far to claim the entire election was “rigged”, like Donald Trump stated multiple times in tweets, he has pushed for every “legal vote” to be counted – insinuating that there were illegal votes cast.
“I Stand With President @realDonaldTrump. We must count every LEGAL vote,” the vice president tweeted on 5 November.
The vice president has not publicly acknowledged Joe Biden as the president-elect, despite it becoming more clear with each passing day that there is no path for Mr Trump to hold a second term in the White House.
Georgia, one of the states Mr Trump has claimed voter fraud occurred, hand recounted every ballot cast. The recount, which finished on Thursday, confirmed that Mr Biden won the state. His lead currently stands at 306 Electoral College votes to Mr Trump’s 232.
Mr Conway, who co-founded The Lincoln Project, has been a vocal opponent of the president and his administration in recent years, even though his wife, Kellyanne Conway, previously worked for the administration.
The vast majority of the Republican Party has yet to speak out against Mr Trump for refusing to accept the election results. Instead, reports have indicated that members of Congress are acknowledging the results privately but placating the president publicly.
Senator Mitt Romney of Utah was one of the few to publicly call Mr Biden the president-elect and condemn the president for promoting unproven voter fraud conspiracy theories.
“Having failed to make even a plausible case of widespread fraud or conspiracy before any court of law, the President has now resorted to overt pressure on state and local officials to subvert the will of the people and overturn the election,” Mr Romney said in a statement on Thursday.
“It is difficult to imagine a worse, or more undemocratic action by a sitting American President.”