James, 28, became the youngest player in NBA history to win four MVPs after receiving 120 of 121 first-place votes by league media members and analysts. The lone other first-place vote went to Knicks star Carmelo Anthony.
"LeBron gets it, I take my hat off to him, he deserves it," said Anthony. "They had a helluva year, to win 27 straight games, a hell of an achievement."
It was the second time a player fell one vote short of an unanimous MVP selection, with Shaquille O'Neal also receiving 120 of 121 first-place votes in 1999-2000.
"It's probably a writer out of New York that didn't give me the vote," James said during Sunday's award ceremony at AmericanAirlines Arena. "I know the history between the Heat and the Knicks. So I get it."
A league spokesman said Sunday the NBA does not release the names of award voters or their ballots, but Gary Washburn of The Boston Globe in an article published in the newspaper Monday acknowledged he cast the sole vote for Anthony.
But coming up short of a unanimous selection didn't overshadow a historic feat for James, who joined Michael Jordan, Bill Russell, Kareem Abdul-Jabbar and Wilt Chamberlain as the only players in NBA history to win at least four MVP awards.
James and Russell are the only players to win the award four times in five seasons. And it was the third MVP trophy James has collected in the span of 12 months. After winning the MVP last season in May, James said he would trade in his third Maurice Podoloff trophy for a first championship. Two months later, he led the Heat to a title and was named Finals MVP after beating Oklahoma City.
The Globe's Washburn said he believed Anthony "meant more to his team" this season.
Washburn wrote that Anthony made the Knicks relevant again by leading them to their first division title in nearly two decades. According to Washburn, the Knicks would not have made the playoffs without Anthony, while the Heat have plenty of other talent besides James. Washburn also wrote that he believed James will eventually become the league's first seven-time MVP and that his vote was "no LeBron conspiracy."
James is coming off the most efficient season he's ever had after shooting a career-high 56.5 percent from the field and 40.6 percent from 3-point range. He averaged 26.8 points, a career-high 8.0 rebounds and 7.3 assists per game.
Now James has his sights set on another evasive honor: Defensive Player of the Year. James was upset two weeks ago when he finished second for the second time in his career for that distinction. Memphis Grizzlies center Marc Gasol was named Defensive Player of the Year.
"I tried harder to win Defensive Player of the Year than I did MVP," James said. "That's my goal. Coach (Erik Spoelstra) had me guarding every position. We'll try again next year."
But before that, James will try help the Heat defend their title when they open Game 1 of their second-round series against the Chicago Bulls on Monday. The Heat haven't played since they completed a four-game sweep of the Milwaukee Bucks on April 28 in the opening round.
"I've been anxious for a while," James said Sunday. "But I'm just excited to get back to basketball right now."
During the ceremony, James acknowledged his fiancée and their two sons, his parents and a group of children who traveled to Miami from his youth foundation in Ohio. The Heat showed a video tribute to James that included jokes and congratulations from teammates and front-office staff.
James laughed during a portion of the video when forward Mike Miller said the two of them now have four MVPs between them, "so let's keep it going."
But there were also emotional moments for James during the video, especially when guard Dwyane Wade told the story of when the two first spent significant time getting to know one another at the 2003 pre-draft camps.
James was the first pick of the 2003 draft, with Heat center Chris Bosh going fourth to Toronto and Wade fifth to Miami.
"God gave him the ability to be a very, very special player," Wade said of James. "You never know how long you're going to have that, and he never takes it for granted."
Perhaps the highest praise for James came from Heat president Pat Riley, who started the ceremony by saying he's been around the NBA game for nearly 50 years as a player, coach, administrator or front-office executive.
During that career span, Riley has played or coached with or against every other member of the four-MVP fraternity James joined with Sunday's MVP announcement.
"Over these 46 years, I've had an opportunity to see some great players -- and all the ones I've observed, watched and have seen, they've always gotten better," Riley said from the podium as he turned to James. "In my humble opinion, I believe the man right here is the best of them all."
Information from ESPNNewYork.com special contributor Ian Begley and The Associated Press was used in this report.