WHITE PLAINS, N.Y.—Democratic presidential nominee Hillary Clinton said she was feeling much better after a pneumonia diagnosis and promised to release additional medical records this week, moving to contain concerns about her well-being and forthrightness after she stumbled exiting a 9/11 ceremony.

The pneumonia diagnosis, belatedly disclosed by her campaign Sunday, has taken Mrs. Clinton off the road and off-message just as her campaign was working to focus on her agenda. Instead of campaigning and fund-raising in California, Mrs. Clinton was resting Monday at her home in New York’s Westchester County and reassuring supporters she is on the mend.

Mrs. Clinton told CNN Monday that she had felt dizzy and dehydrated at the memorial service but quickly felt better once she got into the air-conditioned van. “I’m now taking my doctor’s advice, which was given to me on Friday, that I ignored, to just take some time to get over pneumonia completely,” she said.

Top Clinton officials acknowledged earlier Monday that the campaign made mistakes handling Mrs. Clinton’s exit on Sunday. Press secretary Brian Fallon said the staff had erred in failing to provide information for 90 minutes after Mrs. Clinton’s early departure from the memorial event in lower Manhattan, even as video clips circulated on social media showing her slumping as she tried to step into a van.

When the campaign did address the matter Sunday, a spokesman attributed her early departure to feeling overheated. Only later in the day did the campaign disclose she had been diagnosed with pneumonia on Friday. Mr. Fallon said the campaign wanted the information about Mrs. Clinton’s illness to come from her doctor, which he said wasn’t possible until she examined the candidate again.

All this prompted criticism, even from some supporters, who worry that the incident could fuel some voter concerns that Mrs. Clinton isn’t straightforward.

“Antibiotics can take care of pneumonia,” former Obama adviser David Axelrod wrote on Twitter. “What’s the cure for an unhealthy penchant for privacy that repeatedly creates unnecessary problems?”

Asked why she didn’t disclose more, earlier, Mrs. Clinton, 68 years old, told CNN that she didn’t think “it was going to be that big a deal” and that busy, active people just “keep moving forward” even when they don’t feel well.

She added that she has already disclosed an enormous amount of personal information, including nearly four decades’ worth of tax returns, a summary of her medical records and thousands of her emails as secretary of state. Still, the use of a personal email server for official business has dogged her candidacy.

The campaign didn’t provide specifics on the new medical information that it planned to release.

Mrs. Clinton also challenged Mr. Trump to meet her level of disclosure.

“Compare everything you know about me with my opponent,” she said. She said the campaign takes responsibility for not putting information out quickly over the weekend. “But the information is out there. You can’t say the same thing about Donald Trump.”

On Monday, Republican nominee Donald Trump, 70, was restrained in comments on Mrs. Clinton’s health, instead attacking her recent characterization of half of his supporters as “deplorables.” In an interview on Fox News, Mr. Trump wished Mrs. Clinton a speedy recovery.

On CNN, Mike Pence, Mr. Trump’s running mate, said both presidential candidates should release detailed information about their health. “People are vying for the highest office in the land,” he said. “People have a right to know.”

Mr. Trump said he would put out more medical information, though his timetable wasn’t clear. In December, his campaign released a four-paragraph doctor’s note, which included some health data and assertions such as that his lab results were “astonishingly excellent.”

Mr. Trump hasn’t released any of his tax returns, as nominees have done for four decades. Mr. Pence released 10 years of returns on Friday.

Last year, Mrs. Clinton released the summary of her health. That doctor’s letter outlined past health issues such as deep-vein thrombosis in 1998 and 2009, an elbow fracture in 2009 and a concussion, which was well documented in 2012.

With Mrs. Clinton off the trail, the campaign was relying in part on its deep bench of surrogates. Her husband, former President Bill Clinton, was stepping in to attend Los Angeles fundraisers in her stead, as well as an event Wednesday in Nevada. In addition, events were already scheduled this week for President Barack Obama, first lady Michelle Obama, Vice President Joe Biden and daughter Chelsea Clinton, as well as vice presidential nominee Tim Kaine and his wife, Anne Holton.

Mrs. Clinton, who maintained a full travel schedule as secretary of state and now as a presidential candidate, called in to a San Francisco fundraiser from home on Monday.

Mr. Fallon said Mrs. Clinton hadn’t planned on changing her schedule after her diagnosis of pneumonia on Friday and therefore saw no reason to alert the public of her condition. That day, she attended a national-security meeting, talked to reporters and attended a fundraiser. She participated in meetings on Saturday and didn’t want to miss the 9/11 event the next day, he said.

“Her intention was to power through,” he said. “She was full-steam ahead.” He said that even after Sunday’s incident, aides had to persuade her to cancel the California trip.

Some of her supporters said the incident simply reinforced her strength.

“The woman is super human but still human and last I checked, we all get a little sick sometimes,” Jamie Smith, who traveled as a press aide on her 2008 campaign, wrote in an online essay. “So let’s be clear—what would sideline me (and all of you) for weeks will have her up and running in no time—of that I am certain.”

Mrs. Clinton’s campaign had hoped to discuss her college affordability plan on Monday, but it seemed clear that the health matter was overshadowing that.

“Hillary’s health is good. I’ve had pneumonia. I don’t know about any of you. I’ve had walking pneumonia,” Mr. Biden said from the campaign trail in North Carolina. “What you do, you take antibiotics and you rest a little bit.”

Hillary Clinton left a 9/11 ceremony in New York abruptly Sunday. After videos surfaced showing the Democratic presidential nominee stumble as aides helped lift her to a campaign car, Clinton's doctor released a statement saying the 68-year old had been diagnosed with pneumonia Friday. Credit: Twitter/Zdenek Gazda via Storyful

Dan Pfeiffer, a former Obama adviser, predicted that the issue would quickly fade. “Candidates get sick all the time and work through it,” he said. “The press and public rightly demand transparency on health issues to determine fitness for office. Illnesses like pneumonia have no bearing on that question.”

Write to Laura Meckler at laura.meckler@wsj.com