Martinez won Ajax's player of the season award for the 2021-22 campaign
United manager Erik ten Hag was keen to be reunited with the 24-year-old, who he signed for Ajax in 2019.
He joins the club on a five-year contract which keeps him at Old Trafford until 2027, with the option of a further year.
"It's an honour to join this great football club," Martinez said.
"I've worked so hard to get to this moment and, now that I'm here, I'm going to push myself even further."
Martinez made 120 appearances for Ajax after moving from Argentinian side Defensa y Justicia for £6.3m in 2019, and also played for Newell's Old Boys.
He has won seven caps for Argentina since making his senior debut in March 2019, and played once in the country's successful Copa America campaign in 2021.
"I've been lucky enough to be part of successful teams in my career and that's what I want to continue at Manchester United," he said.
"There will be a lot of work to get to that moment, but I firmly believe that, under this manager and coaches, and together with my new team-mates, we can do it."
Martinez played under Ten Hag at Ajax and won the Eredivisie title with the club in 2021 and 2022, as well as the Dutch Cup in 2021.
He was also Ajax's player of the year for the 2021-22 season.
"He's a warrior and I think the fans will admire him. He has an attitude, fighting spirit," Ten Hag said.
"He brings aggressiveness in the game, in a good way. I think we need that.
"But he's also skilful, he can deal with the ball and he's left-footed."
United's football director John Murtough added: "We are delighted that he has chosen to join Manchester United and we are looking forward to seeing him develop further and help the team achieve the success we are aiming for."
The centre-back is United's third signing of the summer following the arrivals of Christian Eriksen on a free transfer and Dutch full-back Tyrell Malacia.
Efforts continue to lure Frenkie de Jong to Old Trafford from Barcelona.
-- Courtesy of BBC Sport