WASHINGTON—President Donald Trump’s aggressive White House debut is stoking a war with Democrats and creating unease with fellow Republicans, dimming chances for cross-party compromise and potentially limiting the scope of what he can get done while in office.

Democrats, pushed by their base, are under pressure to not cooperate with the new president—on anything. On Tuesday Senate Democrats boycotted committee votes on cabinet nominees and delayed at least for a day a committee vote on Mr. Trump’s choice for attorney general.

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