American climber Brad Gobright, 31, was abseiling down a cliff with Aidan Jacobson, 26, in El Potrero Chico, a popular climbing destination.
Both men fell as they made their descent, but Mr Jacobson escaped with an ankle injury while Mr Gobright fell to his death.
The US State Department confirmed Mr Gobright had died and offered its "sincerest condolences" to his family.
"We are closely monitoring local authorities' investigation and are providing all appropriate consular assistance. Out of respect to the family during this difficult time, we have no further comment," it said.
The climbers had been taking the mountain's Sendero Luminoso, or Shining Path route, which reaches 850 to 900m high.
They were using the technique of simul-abseiling, by which two climbers descend opposite strands of a rope and act as counterweights to each other.
According to Outside magazine, the men had not tied knots in the ends of their ropes, which could have been potentially life-saving.
Many climbers avoid tying "stopper knots" as they can cause ropes to get stuck.
Mr Jacobson told the magazine that the accident was "a blur".
"I was a bit above him," he said. "I was on the left. He was on the right. Then all of a sudden, I felt a pop, and we started dropping.
"He screamed. I screamed. I went through some vegetation, and then all I remember is seeing his blue Gramicci shirt bounce over the edge."
Mr Jacobson said his fall had been broken by a bush before he struck a ledge.
Two Costa Rican climbers who had been above the two men witnessed the accident and helped Mr Jacobson to the bottom of the cliff.
Mr Gobright was a highly experienced climber and had once held the speed record on the Nose of El Capitan.
The publication Rock And Ice described him as "one of the most accomplished free solo climbers in the world".
Many people shared tributes to the young American on social media.
One of his climbing partners, Alice Hafer, wrote: "He had a magic about him on the rock, unlike anyone I've ever met.
"He was so supportive and encouraging, always pushing me harder and believing in me. I can't believe that not even a few weeks ago he was sitting next to me as we drove home from Arizona.
"I'll cherish those moments always. He will be so missed, forever. Love you always Brad."
Another climbing partner, Maison Deschamps, wrote: "Brad climbed for himself because that's who Brad was, a climber.
"He didn't have to talk to people, he didn't need the best sponsors, all he needed was the freedom of the climb. Rest In Peace Brad. I will forever remember your hard work and bravery."
Mr Gobright's death comes after the fall of another high-profile climber, Emily Harrington.
Ms Harrington was taken to hospital earlier this week while trying to scale Yosemite Park's El Capitan and shared her injuries in an Instagram post.