In the footage, Graham argues that if a judge seat becomes vacant during an election year, a nomination should only be considered after a new presidential election has taken place.
Chair of the Senate Judiciary Committee Lindsey Graham has come under pressure after a 2016 video of him reemerged on social media.
The speech took place at a Senate meeting where Republicans blocked the nomination of Merrick Garland, chosen by Barack Obama as a replacement for former Supreme Court Justice Antonin Scalia, who died in the year of the 2016 presidential race.
The video of Graham has gone viral in the wake of the death of Supreme Court Justice Ginsburg, which prompted President Donald Trump to use the opportunity to nominate a Republican figure for the post while he is still in office.
Trump told reporters on Saturday that he will be putting forward the nominee next week, adding that it will be a woman.
Most Democrats have said that the next Supreme Court justice should not be nominated until the next president is installed, echoing 2020 Democratic candidate Joe Biden and former president Barack Obama, who cited the 2016 refusal to consider nominating a new justice in the course of the election campaign.
On Friday, Ginsburg passed away at the age of 87 due to cancer complications at her home in Washington. According to media reports, she told her granddaughter shortly before her death that she wished for her seat to remain vacant until after the next presidential election takes place.