WASHINGTON — Democrats who once saw health care and abortion as their best lines of attack against Judge Brett M. Kavanaugh, President Trump’s Supreme Court nominee, are recalibrating their approach to go after him for his view that a sitting president should not have to answer questions in a criminal case, much less face indictment.

Senator Chuck Schumer of New York, the Democratic leader, said in an interview on Wednesday that Judge Kavanaugh’s belief in broad presidential authority was “just off the deep end.”

For Democrats facing an uphill struggle to block Judge Kavanaugh’s confirmation, his protective views of the presidency could prove to be a bright red ribbon. Rather than just playing it safe with a broad swath of voters worried about access to health care and abortion, Democrats now see an opportunity to excite their base by fanning fears that the highest court in the land could turn into a bulwark to protect the man appointing its members.

As the judge made the rounds of the Capitol — he paid courtesy calls on Senator John Cornyn of Texas, the No. 2 Republican, and Senator Lindsey Graham, Republican of South Carolina — Mr. Schumer and his Democratic colleagues were gaming out a line of questioning that, they hope, will create a perception that a Justice Kavanaugh would compromise the independence of the court.