Pence said the order fulfilled a campaign pledge by Mr. Trump to "implement extreme vetting" on immigrants attempting to enter the country.
If confirmed, Gorsuch would replace one of the court's most stalwart conservatives, Antonin Scalia, who died past year.
Others, however, argue that Democrats should take a page from Republicans' playbook and obstruct Trump and his party at every turn.
Leading Democrats have come out in staunch opposition to Donald Trump's nomination of Neil Gorsuch for the vacant position on the Supreme Court.
Iowa Sen. Chuck Grassley, who chairs the Judiciary Committee that will hold confirmation hearings, declined to say Wednesday whether he would support a move to eliminate the possibility of a filibuster.
Outlining his legal philosophy, Gorsuch said: "It is the rule of judges to apply, not alter, the work of the people's representatives".
Sen. Dianne Feinstein, D-Calif., talks to Sen. Republicans in the Senate should use whatever means necessary to get the job done. Jeff Sessions, R-Ala., Jan. 31, 2017. If Gorsuch fills the spot, the Supreme Court's balance won't change.
"This is a stolen seat being filled by an illegitimate and extreme nominee, and I will do everything in my power to stand up against this assault on the court", said Sen. "He may be more conservative than Kennedy when it comes to expanding individual rights, but he seems to lack Scalia's fervor for overturning liberal precedents from decades past". From here, federal district court rulings overnight Saturday are likely to be appealed, spurring a rolling series of legal debates about one of the first actions taken by President Trump.More news: Congress Threatens Legislation to Keep Russia Sanctions
Kagan is the youngest member of the Court and has established a mostly liberal record in her short time. That puts him well within the Republican mainstream but does not make him an entirely safe bet for Trump. He also spoke of his admiration for Scalia, calling him a "lion of the law".
Meet Trump Supreme Court nominee Neil Gorsuch Neil Gorsuch is now a judge on the 10th Circuit Court of Appeals, but there's already talk of a possible filibuster.
FILE - Sen. Ted Cruz, R-Texas, right, arrives on the West Front of the U.S. Capitol on January 20, 2017, in Washington.
"Neil Gorsuch will make an outstanding Supreme Court Justice". In that case, the nomination would need 60 votes rather than a simple majority.
"The Democrats will not succeed in filibustering this nominee", he told VOA's Kurdish service. I don't know the answer to that.
Earlier, Republicans pushed two of Mr Trump's proposed appointments through a Senate committee vote by suspending the panel's rules. Although Republicans now have a majority in the Senate (52-46 with two Democratic supporting Independents), Democrats are said to be considering a filibuster, which would require a 2/3rds majority (60 votes in the 100-member Senate) to override.
Such an examination could reveal that Gorsuch, like he showed during a gracious speech thanking the president for his nomination, is not the man Democrats want to fight-to quote a frequent phrase of Schumer's-"tooth and nail". "Both got up-or-down votes".