The 22-year-old had hoped to be fit for the April 16 date after sustaining the injury on England Under-21 duty last week but he is expected to be out of action for between four and six weeks.
Richards told www.mcfc.co.uk: "It is not what I wanted to hear at all. I felt sick when I was told. I knew from experience when I did it that it felt like a bad one but I was hoping for the best."
He added: "All I can do now is concentrate on getting back as quickly as possible. I must admit I did think 'why me?' for a while when I heard the news.
"Of all the games to miss, a Wembley date with United is the last one any player would choose."