Greenwood was suspended by United in January 2022 when accusations against him surfaced on social media.
All criminal charges against him were dropped in February, with the Crown Prosecution Service saying it was due to "the withdrawal of key witnesses and new material that came to light".
In a statement to British media, Greenwood said he had "made mistakes" but added: "I did not do the things I was accused of.
After announcing the loan deal with Getafe on Friday, United said in a statement: "The move enables Greenwood to begin to rebuild his career away from Manchester United.
"The club will continue to offer its support to Mason and his family during this period of transition."