Gorsuch is currently a judge on the US Court of Appeals for the Tenth Circuit and, if confirmed by the Senate, would return the court to a conservative majority.
He will fill the seat left vacant following the death of Justice Antonin Scalia in 2016.
Gorsuch described his appointment as a “most solemn assignment”.
He said: “Standing here, in a house of history, and acutely aware of my own imperfections, I pledge that if I am confirmed, I will do all my powers permit to be a faithful servant of the constitutional laws of this country.”
Trump said that Gorsuch’s nomination was another promise kept to the American people.
He said Gorsuch had “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.”