If confirmed by the Senate, the 49-year-old would restore the court’s conservative majority, lost with the death of Justice Antonin Scalia.
The Senate Democratic leader has said he has “very serious doubts” about Judge Gorsuch’s nomination.
The court has the final legal word on many of the most sensitive US issues, from abortion to gender to gun control.
Mr Trump said Judge Gorsuch had a “superb intellect, an unparalleled legal education, and a commitment to interpreting the Constitution according to text”.
“Judge Gorsuch has outstanding legal skills, a brilliant mind, tremendous discipline, and has earned bipartisan support,” Mr Trump said.
He was picked from a shortlist of 21 choices, which Mr Trump made public during the election campaign.
Accepting the nomination, he said: “”It is the role of judges to apply, not alter, the work of the people’s representatives. A judge who likes every outcome he reaches is very likely a bad judge, stretching for results he prefers rather than those the law demands.”