Martinez limped off in the second half after attempting to shield the ball from Hammers defender Vladimir Coufal.
The 26-year-old was playing in only his third league game after more than three months out with a foot injury.
The club said he “is expected to be out of action for at least eight weeks”.
However he will not require surgery.
“We all wish Licha a speedy recovery and look forward to seeing him back on the pitch later in the season,” a club statement added.
United boss Erik ten Hag said after Sunday’s victory that the injury “doesn’t look good” and described it as a “personal disaster” for Martinez.
Martinez missed the latter stages of the 2022-23 season after fracturing a metatarsal in April and suffered “an aggravation” to the injury at the beginning of this campaign.
He missed 22 games in total with his foot injury before making returning against Tottenham on 14 January.
