Since becoming an evangelical Christian in college, Laura Robinson has also called herself a Republican. But after watching the video of GOP presidential candidate Donald Trump making lewd comments about groping women, she took to Facebook to announce that she couldn’t vote Republican this year.

“How can we support someone who’s so derogatory towards women?” Ms. Robinson, a 32-year-old mother of three from Raleigh, N.C., said in an interview.

Mr....