The 24-year-old centre back, who has seven international caps, had spent the past few seasons working under new United boss Erik ten Hag at Ajax.
“I was with him three years at Ajax and I know his style of playing,” Martinez said.
“With him, I improved a lot, to be honest. Hes here in Manchester now, so again the opportunity was there and for sure hes very important in my [career].”
He added: “Its an honour to join this great football club. Ive worked so hard to get to this moment and, now that Im here, Im going to push myself even further.
“Ive been lucky enough to be part of successful teams in my career and thats 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 team-mates, 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.”
Martinez joins Christian Eriksen and Tyrell Malacia as United’s key off-season signings so far, although rumours continue to swirl about Cristiano Ronaldo’s future at the club.
The Red Devils have seen Edinson Cavani, Jesse Lingard, Juan Mata, Nemanja Matic, Andreas Pereira and Paul Pogba all depart this off-season.