Amid concerns from some family members -- and attacks from Democrats -- that his independent campaign could swing a close race between Donald Trump and Joe Biden, Robert F. Kennedy Jr. is continuing to push ahead to gain access to ballots around the country, which will ultimately determine his success with voters.

Kennedy said Monday in a a CNN interview that he would have "no problem" getting on the ballot in every state, including several key battleground states where Biden and Trump's small margins of victory in previous elections could be whittled away by Kennedy. Only one state has confirmed he is on the ballot: Utah.

Kennedy is running for president as an independent candidate, which means he is not guaranteed to a spot on any state's ballot. That means his campaign is working to gather signatures -- anywhere from about 900 to 42,000 required signatures -- to get on states' ballots.

Public polling indicates Kennedy is a long shot to win the presidency. He scratches double-digits in some recent general election polls, but that's it. In the early phase of the election, surveys have indicated some interest in third-party options compared with Biden or Trump, however. An ABC News/Ipsos poll conducted in early January found that 37% of adults said they would be somewhat or very likely to seriously consider a third-party option with Biden and Trump as the major party nominees.

With a rematch now set between Biden and Trump in November, despite what polling shows are low approval ratings for both men, Kennedy's independent campaign has attracted more notice -- and increasing criticism from Democrats that he's nothing more than a "spoiler" who will siphon off enough Biden-leaning voters to ensure Trump's reelection.

He rebuffed that worry during his CNN interview, describing his White House bid as unique from either Biden or Trump.

On Monday, he rejected the idea that he selected Silicon Valley lawyer Nicole Shanahan as his running mate because she is wealthy enough to help fund his effort to access every state's ballot.

"We don't need her money to get on the ballot in every state," he said of Shanahan, who has donated millions of dollars to a pro-Kennedy Super PAC, during a CNN interview.

While only Utah has confirmed Kennedy is on the ballot, Kennedy and the PAC supporting him say they have enough signatures to get him on the ballot in eight other states. He still has enough time to get on many more ballots if his campaign or the super PAC get enough signatures.

His campaign says they have gotten enough signatures to get on the ballot in four states as either an independent candidate or as registering the "We the People" party, the party formed by Kennedy: Nevada, North Carolina, New Hampshire and Hawaii.

In Nevada, Kennedy's team failed to name a running mate when it filed its petition to get him on the state's ballot, which Nevada statutes require of independent candidates, according to documents filed by Kennedy's campaign and shared with ABC News by the Nevada Secretary of State's Office. The campaign claims this is an attempt to invalidate the candidate. The office has yet to announce how it is moving forward and whether Kennedy could be added.

In North Carolina, the state board of elections said Kennedy has until May 17 to submit signatures to county boards, and then until June 3 to submit to the state board of elections. As of April 1 the state board said it had not received any signatures.

In 2020, Trump won North Carolina by around 74,000 votes -- about a 1% margin over Biden.

In New Hampshire, the secretary of state's office doesn't start counting signatures until September.

In Hawaii, the We the People party qualification petition is facing an objection from the Democratic Party of Hawaii, with a hearing to be held on April 5.

The super PAC supporting Kennedy, American Values 2024, said it received enough signatures to get Kennedy on the ballot in four states: Georgia, Arizona, Michigan and South Carolina.

In Georgia, petitions can only be filed starting in June.

In Arizona, the candidate filing period opens on July 28.

In 2020, Biden won Arizona by just 10,457 votes, and Georgia by just 11,779 votes.

In Michigan, as of March 12, the secretary of state's office has not gotten any petitions from RFK; he has until July 18 to turn them in.

In South Carolina, the South Carolina Elections Commission received a petition on March 21.