The Dutch defender leaves Ajax Amsterdam after three seasons at the Johan Cruyff Arena, where he won two Eredivisie titles and the Dutch Cup.
Martinez has signed a five-year contract at Old Trafford with the option to extend for a further 12 months.
The 24-year-old is Manchester United's third summer signing, joining Tyrell Malacia and Christian Eriksen at the club.
It was Manchester United manager Erik Ten Hag who signed Martinez for Ajax from Argentina side Defensa y Justicia in the summer of 2019 and the pair will get the chance to work together again in England.
"It's an honour to join this great football club. 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 told the club's official website.
"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. 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 teammates, we can do it.
"I want to thank Ajax and their fans for all the support they gave me. I had an unbelievable time there but feel the moment is right to test myself in another environment. Now I am at the perfect club to do this."
Ten Hag added: "He's a warrior and I think the fans will admire him. He has an attitude, fighting spirit. He brings aggressiveness in the game, in a good way. I think we need that. But also, he's also skilful and he can deal with the ball and he's left-footed."
"I think the Premier League is high intensity, lots of challenges. We need such players. I know from South Americans... I call it grinta. Controlled, aggressive. You need that in your game and with him we have such a player."
Martinez has won seven caps for the Argentina national team and was part of the side that won the Copa America last year.
