WASHINGTON—More than half the Democrats in the Senate now say they will vote against President Donald Trump’s nominee to the Supreme Court, setting up an acrimonious showdown over Judge Neil Gorsuch’s bid to fill the seat left vacant by Antonin Scalia’s death last year.

The battle shows signs of becoming bitter, with Democrats promising a filibuster, Republicans threatening to invoke a rules change known as the “nuclear option,” and Democrats essentially daring them to try.

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