Mr. Elvig said he had been calling the Boy Scouts office for hours on Tuesday to express his dismay, but had been getting a busy signal.
“I would like a public denouncement of what happened yesterday and reaffirmation of the values I think I learned in Scouts,” Mr. Elvig said. “If they can’t do that, I will be returning my medal.”
Either way, the firestorm was an unwelcome and surprising development during a gathering that is among Scouting’s most important events, a quadrennial meeting that attracts tens of thousands of people and, very often, presidents, who in the past have spoken about service, values or citizenship, not partisan politics.
Mr. Trump’s appearance before an enthusiastic crowd of neckerchief-clad, saluting Scouts at a 14,000-acre compound was a distinct break from 80 years of presidential speeches to the nation’s Scouts.
In the speech’s opening moments, it seemed that Mr. Trump, who was not a Boy Scout as a youth, would mostly avoid talking about the partisan clashes that have divided Washington.
“I said, who the hell wants to speak about politics when I’m in front of the Boy Scouts? Right?” Mr. Trump said shortly before he extolled the Scouts as “young people of character and integrity who will serve as leaders in our communities, and uphold the sacred values of our nation.”
But the speech by Mr. Trump, the 19th occupant of the White House to also serve as the honorary president of the Boy Scouts of America, was ultimately punctuated by a brand of political oratory that proved startling at a Boy Scout gathering.
He recounted how he won last year’s presidential election: “We won Florida. We won South Carolina. We won North Carolina. We won Pennsylvania.” He said Hillary Clinton “didn’t work hard” in Michigan, a state Mr. Trump won, and he resurfaced his grievances with “fake news” and “fake polls.” And when he landed on the second point of the Scout Law — loyalty — Mr. Trump interrupted himself to say, “We could use some more loyalty, I will tell you that.”
Presidents of both parties have been connected to the Boy Scouts: Their signatures have been affixed to Eagle Scout certificates, they have hosted boys and leaders in the Oval Office and many have appeared at jamborees.
In 2005, George W. Bush reminded the Scouts that Franklin Delano Roosevelt had appeared at a jamboree in 1937, and he spoke about themes that are familiar to Scouts, including service and character.
“When you follow your conscience, and the ideals you have sworn as a Scout, there is no limit to what you can achieve for our country,” Mr. Bush said.
Bill Clinton, who spoke in 1997, made similar comments.
“We need you to remain focused on the strong values you learned in Scouting, to remember that character counts and service counts,” Mr. Clinton said. “We need you if we’re going to build our communities and bring our people together across all the lines that divide us.”
The organization has faced frustration and anger in recent decades for its policies on gay and transgender people, and the issue even reached the United States Supreme Court in 2000. Although the Scouts won that case, which involved the organization’s expulsion of an openly gay adult leader, the group has struggled to cultivate cultural relevance and stem a collapse in membership.
The group said this year that it had more than 2.3 million youth participants. About a decade earlier, it had close to 2.9 million participants.
The group has also faced separate criticism and litigation for its handling of sexual misconduct, which is documented in files that the organization has closely held. In 2010, for example, an Oregon jury said the Scouts should pay $19.9 million, including $18.5 million in punitive damages, to a man who had been abused by an adult leader.
But some said in this case, it was not the Scouts who were at fault.
Zach Wahls, a co-founder of Scouts for Equality, which pressured the Boy Scouts to allow gay and transgender members, said Mr. Trump’s speech put the Scouts “in a very difficult position that they didn’t want to be in.”
“The Boy Scouts were not in the wrong here,” said Mr. Wahls, 26, who became an Eagle Scout while growing up in Iowa. “We should not be blaming the organization that always invites the president to speak. We should be talking about the president who took that opportunity and twisted it.”