WASHINGTON—Race has become a flashpoint in the growing dispute between House Speaker Nancy Pelosi and a quartet of liberal freshmen lawmakers, again pitting Democratic members against each other in starkly personal terms.

Rep. Alexandria Ocasio-Cortez (D., N.Y.) told the Washington Post that she felt Mrs. Pelosi was persistently singling out “newly elected women of color” and the speaker’s comments were disrespectful. The remarks drew fire on Thursday from fellow members of color in the caucus.

“What...