On Twitter, the Avengers star writes that despite continued brain fog, he is happy to be able to watch the premiere of season two of "Mayor of Kingstown" at home with his family.
It was when Renner was plowing a road at his ranch in Nevada that the accident occurred.
On social media, Renner has previously shown how he tackles the snow masses on the property with the help of a piste machine equipped with a plow.
The area was hit by severe weather over the New Year's weekend, and somehow the machine came rolling and Renner was partially run over.
His condition was described as critical but stable when he was taken to hospital, and Renner has since been treated in intensive care for chest and leg injuries.
51-year-old Renner is best known for his role in the war drama "The Hurt Locker", for which he was nominated for an Oscar, and as Hawkeye in the Marvel superhero films.
He has one of the main roles in the thriller series "Mayor of Kingstown", whose second season has just started on Paramount+.