The 21-year-old was informed by the Red Devils that he would not form part of their plans for 2023-24 following a long-running internal investigation.
He has not played a competitive game since January 2022.
Charges of attempted rape, controlling behaviour and assault were dropped in February 2023, allowing Greenwood to resume his professional playing career.
United invited offers for the one-cap England international, with interest reportedly shown from Italy and Turkey.
A deal was eventually done that has taken Greenwood to Spain, with a fresh start being sought outside of England.
Greenwood has told the English-language Twitter account set up by his new club: “Hi Getafe fans, it's Mason here. I'm so happy to be here and I can't wait to get started.”
Getafe coach Jose Bordales has said of welcoming the fleet-footed winger into his squad: "It is too delicate a situation to trivialise this issue. Everybody knows what happened, the appropriate measures were taken.
"Obviously we can only talk about football, about other issues, I think, that the people and the relevant systems did what they had to do, and everybody knows how it ended - without a condemnatory sentence.
"He's a footballer of the highest level, who comes to Getafe with enormous hope. We are going to help him to recover his best level."
Greenwood has the international break to continue working on his fitness, with it possible that he could make his debut for Getafe when they face Osasuna on September 17.