BERLIN—The center-left candidate for chancellor in Germany’s September election demanded that U.S. nuclear weapons be withdrawn from the country, taking an increasingly anti-American tack in a campaign season overshadowed by the European discontent with President Donald Trump.

Martin Schulz of the Social Democrats, the challenger to incumbent Chancellor Angela Merkel, said Wednesday that the “upper limit for nuclear weapons in our country must be zero.” He was elaborating on comments he made at a campaign rally in southwest...