Craft serves as the U.S. ambassador to Canada. In February, Trump had said he had selected Craft for the U.N. job.
Craft, a Kentucky native, was a member of the U.S. delegation to the U.N. General Assembly under President George W. Bush's administration.
Senate Majority Leader Mitch McConnell, R-Ky., backed Craft for the post.
As U.S. ambassador to Canada, she played a role in facilitating the U.S.-Mexico-Canada trade agreement, a revamp of the North American Free Trade Agreement.
Craft has been ambassador during a low point in relations. Last year Trump called Canadian Prime Minister Justin Trudeau weak and dishonest, words that shocked Canadians.