Grant, who is due to reunite with his Strictly Come Dancing partner Flavia Cacace in the Whatsonstage.com Award-nominated show, slipped on ice in Regent's Park last month and has been unable to perform since previews began at the Phoenix Theatre on 30 January.
Cacace and co-star Vincent Simone said in a joint statement: "We are so sorry that our dear friend Russell can't join us on stage to perform on the opening night of Midnight Tango, but we look forward to having him join us in the show as soon as he is able to perform again. We wish him a very speedy recovery."
Producer and former Strictly judge Arlene Phillips added, "We were looking forward to celebrating Russell's birthday with him tonight. Now, sadly, Russell has been injured and won't be able to perform, which I know upsets him just as much as us. I wish him a speedy recovery, so that he can be back on stage, where he belongs, as soon as possible."
Grant, who was paired with Flavia during the 2011 Strictly series, plays Argentinian bar owner Carlos in the show, which runs for a strictly limited season at the Phoenix Theatre from 30 January to 2 March 2013.
The West End run is followed by a UK tour which kicks off in Eastbourne from 13 to 16 March 2013 and will visit Norwich, Woking, Southend, Bath, Nottingham, Wycombe, Birmingham, Newcastle, Truro, Cardiff, Hull, Northampton, Canterbury, Manchester and Leeds.