The Des Moines Register says the billionaire cares more about promoting his brand than helping the country, reports CBS News correspondent Nancy Cordes.
An editorial says, "If he were merely a self-absorbed, b-list celebrity, his unchecked ego could be tolerated as a source of mild amusement. But he now wants to become president."
Iowa is the first key caucus state.
The newspaper's editors didn't mince words, saying Trump is "polluting the political waters and keeping qualified candidates from getting their message out."
Trump drew the ire of many fellow Republicans with some tough words about the military service of Arizona Sen. John McCain, who was a POW during the Vietnam War.
"He's [considered] a war hero because he was captured. I like people who weren't captured," Trump said.
In a fiercely worded editorial, the Register warns that comments like that "threaten to derail not just his campaign but the manner in which we choose our nominees for president."
Trump also said, "Jeb Bush would be a joke on immigration."
Trump's name-calling, the paper says, has turned him into "the distraction with traction - a feckless blowhard who can generate headlines ... not by provoking thought, but by provoking outrage."
"He's a jackass," says South Carolina Sen. Lindsey Graham, a fellow GOP presidential contender.
Another Trump rival for the Republican nomination, former Texas Gov. Rick Perry, weighed in against what he called "the cancer of Trump-ism."
Perry said on Fox News, "Being a disciplined candidate is really important and that is, I think, what his challenge is gonna be."
Trump told CBS News' Charlie Rose his opponents are just taking shots at him to get attention.
"I'm leading the polls," Trump said. "And they've got zero. I mean, one has 1 percent. One has another 1 percent. One has zero.
A new national poll has Trump 11 points ahead of his nearest GOP opponent, though some respondents were surveyed before Trump made his comments about McCain.
"He graduated last in his class at (the U.S. Naval Academy at) Annapolis," Trump also said of McCain.
Republican strategist Rick Davis ran McCain's presidential campaign in 2008. He says there's no question Trump has singlehandedly changed the tone of this race.
"The reality is up until this point in this campaign has been civil, smart. ... If not for Donald Trump, we wouldn't be having this conversation," Davis said.
Trump did finally apologize to McCain Monday night. Sort of. After he was browbeaten into it by Fox News's Bill O'Reilly.
"I want you to say something to Sen. McCain tonight, man to man, right to him right now," O'Reilly said.
"Certainly, if there was a misunderstanding, I would totally take that back, but hopefully, I said it correctly," Trump replied.
And if it wasn't correct, Trump went on to say, he did correct himself shortly thereafter.
The questions this morning are will other editorial boards begin to follow the Register's lead, and will any of it matter to trump?
It's a safe bet he'll have something to say about it at a morning campaign event in Bluffton, South Carolina.