"We are bitterly disappointed for Martyn," said Dragons Rugby Manager Robert Beale.
"He has had a hard road to recovery from his previous injury and this is a heavy blow for him.
"However, he has shown great strength of character through his rehabilitation and we are confident that he will meet this next challenge head-on."
Dragons lock and captain in the absence of Tom Willis, Luke Charteris picked up an injury to his ribs and will undergo further assessment this week.
Wing Aled Brew, who twisted his ankle in the match on Sunday is expected to be back for the region's clash with the Ospreys on Friday 15th April (7:05pm).