President Donald Trump's nominee, Neil Gorsuch, faces a fierce confirmation battle. He needs at least 60 votes from the country's 100 senators to prevent opponents from launching a filibuster - a procedure that essentially prevents a vote through endless debate.
"Make no mistake about it. This would be an unwise and unprecedented act," Pence said of that filibuster threat, during a speech in Philadelphia to a local chapter of the Federalist Society, an influential organisation that promotes a conservative evolution of US law.
Many Democrats deeply resent last year's refusal by Senate Republican leaders to even consider former President Barack Obama's nominee, Merrick Garland, to fill the vacancy. The seat has sat empty since the death, nearly a year ago, of conservative justice Antonin Scalia.
Supporters of Gorsuch cast the 49-year-old federal judge as an ardent defender of the conservative values championed by the late Republican icon.
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With only 52 seats in the Senate, the majority Republicans will have to convince at least eight Democrats to cross party lines to vote for Gorsuch, who supports the death penalty and other core right-wing positions.
If support falls short, Trump has urged Senate Republicans to change longstanding rules - using what is referred to as the "nuclear option" - to allow Gorsuch's confirmation by a simple majority of 51 votes, a highly controversial approach.
Pence's campaign for Gorsuch came as Trump attacked another federal judge who blocked the president's controversial visa bans targeting seven majority-Muslim countries.
US District Judge James Robart of Washington state - an appointee of Republican president George W Bush - issued a nationwide order blocking Trump's bans late Friday.
By early today, the US government began implementing the judge's order.
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