WASHINGTON—President Trump pledged to retaliate against any investigations launched by majority House Democrats in the new Congress, during a post-midterm news conference in which he repeatedly struck a combative stance while also calling for bipartisanship and hailing Republican Senate victories.

If Democrats, when they take the majority, follow through on their pledges to investigate what they say are conflicts of interest facing the White House and the president’s company, Mr. Trump said that would create a “warlike posture.”

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