WASHINGTON—Republican intraparty warfare over Donald Trump spun into the public arena Monday after Speaker Paul Ryan told his House colleagues he wouldn’t defend or campaign with the party’s presidential nominee.

While Mr. Ryan didn’t rescind his endorsement of Mr. Trump, he drew immediate outrage from the nominee and his supporters, who said the speaker risked undermining the candidate just four weeks from Election Day. It came days after the release of a 2005 video showing the GOP nominee talking about kissing and groping...