Announcing his pick in the White House rose garden, Mr Trump called Ms Barrett a "brilliant legal mind" and praised her "towering intellect, sterling credentials and unyielding loyalty to the Constitution."
He called on Democrats to give her a "quick, straightforward and dignified" confirmation hearing in the Senate.
Mr Trump told Ms Barrett: "It should be very easy, good luck, it should be very quick. You are really fantastic."
Ms Barrett, 48, a mother-of-seven, was accompanied by her family at the ceremony.
She said she was "humbled" to be nominated, adding: "I love the United States, and I love the United States Constitution."
Republicans in the Senate indicated they would try to make the confirmation one of the quickest in modern history, holding hearings in the week of Oct 12 and a full vote by Oct 29.
The Senate has never before confirmed a Supreme Court justice beyond July in an election year, and Democrats accused Republicans of being “hell-bent” on doing it without due process.
Supporters of Mr Trump said they expected the confirmation fight to be “jet fuel for our base”.
There are less than 40 days until the election, and the average time for a confirmation process is about 70 days.
An ABC/Washington Post poll showed 57 per cent of Americans thought the choice of a Supreme Court justice should be left to the winner of the election, and only 38 per cent believed it should go ahead before Nov 3.
The confirmation of Ms Barrett, a conservative appeal court judge, could have deep ramifications for issues including abortion and Obamacare.
A case against Obamacare, backed by Mr Trump's administration, is due to come before the court days after the election and the law, officially known as the Affordable Care Act, could fall.
Democrats indicated they would make saving Obamacare a cornerstone of their opposition, both to confirming Mr Trump's nominee, and the election campaign.
Anti-abortion groups have been vocal in support of Ms Barrett, and believe her confirmation could lead to the overturning of Roe v. Wade, the 1973 law legalising abortion.
Republicans said they had the votes needed to confirm her with a simple majority in the Senate
The party holds the Senate 53-47 and only two of its senators have objected to holding a vote before the election.
At age 48 Ms Barrett, who is also a professor at the University of Notre Dame, would be the youngest of the nine Supreme Court justices, who are appointed for life.
That could help stamp Mr Trump's vision on the court for decades.
She would replace Ruth Bader Ginsburg, the liberal justice who died aged 87 on Sept 18, increasing the conservative majority on the court to 6-3.
Mr Trump has said he wants the full nine justices in place by November 3 in case the Supreme Court has to decide a disputed election.
He met with Ms Barrett on Monday and Tuesday at the White House, and did not interview anyone else.
Ms Barrett's confirmation hearing will take place in front of the Senate judiciary committee.
Some Democrats suggested boycotting the hearings, but the party's members on the committee indicated they would show up.
Kamala Harris, the Democratic vice-presidential nominee, sits on the committee and will question Ms Barrett.
Mitch McConnell, the Republican leader in the Senate, said he expected Ms Barrett to be subjected to a "hurricane of misrepresentations".
During her 2017 confirmation hearing for the appeal court Dianne Feinstein, a Democrat senator, said of Ms Barrett, that "the dogma lives loudly within you", which was condemned as an attack on her Catholic faith.
Ms Barrett is considered a Constitutional "textualist," believing that the original intentions of the framers should be adhered to closely in Supreme Court rulings.
She was once nicknamed "The Conenator" by fellow lawyers for her ability to destroy weak legal arguments.
The judge was born in New Orleans and now lives in South Bend, Indiana, near Notre Dame.
Her seven children are all under 20, including one with Down syndrome, and two adopted from Haiti.
She would be the first mother of school-aged children on the Supreme Court.
Her nomination is expected to lead to scrutiny of her reported membership of a non-denominational Christian group called People of Praise.
Until 2018, the group used the term "handmaid" for its female leaders. A decision to change the term was made following a popular TV series based on "The Handmaid’s Tale," the 1985 book by Margaret Atwood, which depicts a dystopian future in which women are subjugated.
People of Praise has emphasised that it does not consider women subservient, and that many hold leadership roles.