Donald J. Trump belittled the parents of a slain Muslim soldier who had strongly denounced Mr. Trump during the Democratic National Convention, saying that the soldier’s father had delivered the entire speech because his mother was not “allowed” to speak.

Mr. Trump’s comments, in an interview with George Stephanopoulos of ABC News that will air on Sunday, drew quick and widespread condemnation and amplified calls for Republican leaders to distance themselves from their presidential nominee. With his implication that the soldier’s mother had not spoken because of female subservience expected in some traditional strains of Islam, his comments also inflamed his hostilities with American Muslims.

Khizr Khan, the soldier’s father, lashed out at Mr. Trump in an interview on Saturday, saying his wife had not spoken at the convention because it was too painful for her to talk about her son’s death.

Mr. Trump, he said, “is devoid of feeling the pain of a mother who has sacrificed her son.”

Gov. John Kasich of Ohio, a rival of Mr. Trump’s in the Republican primaries who has refused to endorse him, castigated him on Twitter. “There’s only one way to talk about Gold Star parents: with honor and respect,” he wrote, using the term for surviving family members of those who died in war.