The winger pulled up in the second half at The Emirates and, despite trying to play on, was forced to depart.
"It looks like a serious hamstring," manager Arsene Wenger said. "First we assessed the hamstring, he wanted to go on and stay on the pitch and I think he should have gone off straight away because the second time it was a real hamstring (injury).
"I think now for him it will be tough to come back before the end of the season. It is usually 21 days (minimum)."
Walcott is expected to be part of England's Euro 2012 squad for Poland and Ukraine and Wenger said he expected his player to recover in time for the June 8-July 1 championship.
"He should be okay to play for England - he just will have a good rest. But I don't think he will be ready to play for us."
Arsenal, who are trying to secure a top-four finish in the Premier League, have three matches left this season.