"I regret that if my referencing that moment of trauma for our entire nation and in particular the Kennedy family was in any way offensive. I certainly had no intention of that whatsoever," the former first lady said.
The episode occurred as Clinton campaigned in advance of the June 3 South Dakota primary.
Responding to a question from the Sioux Falls Argus Leader editorial board about calls for her to drop out of the race, she said: "My husband did not wrap up the nomination in 1992 until he won the California primary somewhere in the middle of June, right? We all remember Bobby Kennedy was assassinated in June in California. You know I just, I don't understand it," she said, dismissing the idea of abandoning the race.
Clinton said she didn't understand why, given this history, some Democrats were calling for her to quit.
Her remark about an assassination during a primary campaign drew a quick response from aides to Democratic presidential front-runner Barack Obama.
"Senator Clinton's statement before the Argus Leader editorial board was unfortunate and has no place in this campaign," said Obama spokesman Bill Burton.
Clinton spokesman Mo Elleithee said the senator was only referring to her husband and Kennedy "as historical examples of the nominating process going well into the summer and any reading into it beyond that would be inaccurate and outrageous."
Even so, Clinton decided within a couple of hours to make a personal apology.
"I was discussing the Democratic primary history and in the course of that discussion mentioned the campaigns of both my husband and Senator (Robert) Kennedy waged in California in June in 1992 and 1968 and I was referencing those to make the point that we have had nomination primary contests that go into June. That's a historic fact," she said.
"The Kennedys have been much on my mind the last days because of Senator Kennedy," she added, referring to Sen. Edward M. Kennedy's recent diagnosis of a brain tumor. "I regret that if my referencing that moment of trauma for our entire nation and in particular the Kennedy family was in any way offensive. I certainly had no intention of that whatsoever.
"My view is that we have to look to the past to our leaders who have inspired us, give us a lot to live up to, and I'm honored to hold Senator Kennedy's seat in the United States Senate from the state of New York and have the highest regard for the Kennedy family," she said.
In the same editorial board meeting, Clinton said her campaign has had no discussions with Obama's aides about her possibly becoming his vice presidential pick.
"It is flatly untrue and it is not anything I'm entertaining. It is nothing I have planned and it is nothing I am prepared to engage in. I am still vigorously campaigning."
The Obama campaign also dismissed reports that there were talks going on between the two campaigns about putting Clinton on the ticket.
Obama has an almost 200-delegate lead over Clinton and is just 56 delegates short of the number needed to clinch the nomination, making Clinton's goal of catching him more difficult by the day. The primaries end June 3.
Clinton spent the day campaigning in South Dakota, which holds one of two June 3 primaries. At stake are 15 delegates.
Recent reports suggested she may be discussing ways to end her campaign by being offered the vice presidential slot underneath Obama, but she rejected that and said she suspected the talk was coming from Obama aides.
"I would look to the camp of my opponent for the source of these stories," she said. "People have been trying to push me out of this ever since Iowa."
Two of those recent reports, however, were attributed by CNN and The New York Times to supporters of Clinton.
Meanwhile, Sen. Dianne Feinstein, D-Calif., a staunch Clinton supporter, said Friday that she believes that if Obama becomes the nominee he should select Clinton as his running mate.
"I think as this race has emerged each one of them has garnered a different constituency and different states, and therefore when you put the two of them together it forms, I believe, the strongest ticket," she told The Associated Press in a phone interview.
"Women feel very strongly about Hillary and African-Americans feel very strongly about Barack, and the election results show that, and the young versus old, the higher educated versus the working person. ... All these things are sort of separated out into one or the other so there is a logic in combining the two constituencies."
Feinstein is a longtime friend and supporter of Clinton's. So would Clinton accept the vice-presidency?
"I think anyone accepts if asked -- whatever they say," Feinstein said.
Former Fannie Mae CEO Jim Johnson is overseeing the early vetting of possible vice presidential running mates for Obama, Democratic officials say. He did the same job for Democratic nominees John Kerry in 2004 and Walter Mondale in 1984.