After receiving criticism from her own party, the representative for Wyoming defended the friendly gesture during Mr Biden’s first address to a joint session of Congress.
"I disagree strongly w/ @JoeBiden policies, but when the President reaches out to greet me in the chamber of the US House of Representatives, I will always respond in a civil, respectful & dignified way," she said in a tweet.
"We’re different political parties. We’re not sworn enemies. We’re Americans."
The third most senior Republican in the House has been fending off calls for her ouster from party leadership, and her House seat, over her vote to impeach Mr Trump following the riot at the US Capitol.
The brief fist bump with Mr Biden on Wednesday renewed that rhetoric among supporters of the former president, with Donald Trump Jr criticising the "warmonger".
"So glad she’s in the GOP leadership, I guess they wanted to be more inclusive and put Democrats in there too?!?" he said in a tweet.
Mr Trump Jr was invoking his father earlier in the week calling Ms Cheney a "warmongering fool" who would embarrass her family by "running for president, in order to save face".
Ms Cheney, daughter of former vice president Dick Cheney, told the New York Post on Monday that she was "not ruling anything out" when asked if she would consider running for president in 2024.
"I do think that some of our candidates who led the charge, particularly the senators who led the unconstitutional charge, not to certify the election, you know, in my view that’s disqualifying,” she told the Post.
That criticism of the GOP has put her at odds with party leaders seeking to downplay its role in events leading up to the riot at the US Capitol on 6 January.
House minority leader Kevin McCarthy refused to say whether Ms Cheney was still a "good fit" for party leadership when asked by Punchbowl News’ Jake Sherman at the party’s annual policy retreat.
"That’s a question for the conference," Mr McCarthy responded. "I think from a perspective, if you’re sitting here at a retreat that’s focused on policy, focused on the future of making America in the next century, and you’re talking about something else, you’re not being productive."
Two Republican lawmakers told CNN that Ms Cheney’s comments this week "opened old wounds" and that she wasn’t speaking for the party, which was more interested in moving on than ousting the Congresswoman from leadership.
"It’s real and much more widespread than before and completely of her own making," the lawmaker told CNN on condition of anonymity. "At this point, it has zero to do with her vote and everything to do with her words and actions."
Senate minority leader Mitch McConnell said this week the GOP was "looking to the future, not the past" in response to Mr Trump calling for new leadership. Mr Trump has backed Mr McCarthy as continuing the former president’s legacy in the 2022 midterms and beyond.
Mr McConnell said after this week’s address to the joint session of Congress that the future of the party was in people like South Carolina senator Tim Scott, who delivered the GOP response to Mr Biden.
"We’re looking to the future, not the past, and if you want to see the future of the Republican Party watch Tim Scott’s response to President Biden last night," Mr McConnell told Fox News. "He’s the future, that’s where we’re headed."
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