WASHINGTON — As the Republican candidates for the White House battled in Wisconsin last week, Speaker Paul D. Ryan was conspicuously absent from his home state — but he was very much on the political stage.

He visited Prime Minister Benjamin Netanyahu in Israel, where he also met with local reporters and made several statements affirming the United States’ commitment there, before heading to other Middle Eastern nations and Germany to discuss security and intelligence issues.

Back in Washington, his staff churned out its latest flattering video of Mr. Ryan, deploring identity politics and promoting a battle of ideas — set to campaign-style music. And his office continued to beat back the not-exactly-library-voice whisper campaign favoring a coup at the Republican convention in July that would elevate Mr. Ryan to the top of the ticket.