Bryant and his daughter Gianna were among nine people killed in a helicopter crash on Sunday.
LA Lakers star James surpassed Bryant to become the NBA’s third-highest scorer of all time on Saturday.
James also played alongside Bryant at the 2008 and 2012 Olympics, helping the United States win two gold medals.
“Man, I sitting here trying to write something for this post but every time I try, I begin crying again just thinking about you, niece Gigi and the friendship/bond/brotherhood we had!” he posted on social media.
“I literally just heard your voice Sunday morning before I left Philly to head back to LA. Didn’t think for one bit in a million years that would be the last conversation we’d have.
“I’m heartbroken and devastated my brother.
“I promise you I’ll continue your legacy man! You mean so much to us all here. Please give me the strength from the heavens above and watch over me! I got US here! There’s so much more I want to say but just can’t right now because I can’t get through it! Until we meet again my brother!!”
Bryant’s final post on Twitter before his death was a message to James, saying: “Continuing to move the game forward KingJames. Much respect my brother.”
The pair never played together in the NBA but James, who joined the Lakers two years after Bryant’s retirement, spoke about how much he looked up to his iconic long-time rival and friend.
The NBA has called off the Los Angeles Lakers’ match against the LA Clippers on Tuesday.
Bryant finished his playing career as the Lakers’ all-time leading points scorer, and is fourth on the NBA’s all-time list with 33,643 points.
In his 1,566th and final game for the Lakers in April 2016 he scored 60 points for the sixth time.
More than one million people have backed a petition to use his image as a new official NBA logo.
Speaking after the LA Lakers’ 108-91 defeat by the Philadelphia 76ers on Saturday, which was the day before Bryant’s death, James said: “I’m happy just to be in any conversation with Kobe Bryant.
“One of the all-time greatest basketball players to ever play, one of the all-time greatest Lakers.”
Super Bowl stars mourn Bryant
Players and coaches paid tribute to Bryant at the Super Bowl’s opening night in the build-up to Sunday’s match between the Kansas City Chiefs and the San Francisco 49ers.
Chiefs quarterback Patrick Mahomes: “I wasn’t lucky enough to meet Kobe but the impact he made in my life was huge.
“The work ethic, that intensity that he had to be great every single day.
“Even to this day, I still watch videos on YouTube the day before games, just listening to him talk, how he puts everything into perspective being great on and off the court.”
49ers cornerback Richard Sherman: “I know how he would want me to take care of business in this game.
“I was sad this morning, down in the dumps. I know he would tell me to quit being a baby. He was a tremendous idol. There’s not enough praise I can give him.”
Chiefs tight end Travis Kelce: “You can’t say enough about who he was and his impact.
“I just feel bad for the Bryant family, everybody involved. My heart’s with you and everyone here in America.”
Chiefs coach Andy Reid: “It’s sad. I knew Kobe from Philly, and he was a great person. I feel bad for his family, sick for his family.
“They’ll rebound. They’re strong. They’ll live up to his strength.”