Arsenal boss Arsene Wenger says the 22-year-old, capped 24 times, will be out for a further six to eight weeks.
Oxlade-Chamberlain suffered his initial knee injury in the Champions League defeat by Barcelona on 23 February.
"There is no chance of the Euros for him," said Wenger. "He will be back at the beginning of July."
Oxlade-Chamberlain, who made his England debut against Norway in 2012, had been scheduled to feature for the Gunners' Under-21 side earlier this week.
"He was out for a while and was not overloaded with games," Wenger added.
"You want a guy of 22 years of age to go to the European Championship, so it is very sad."
Oxlade-Chamberlain was a surprise call-up to Roy Hodgson's squad for Euro 2012, becoming the third-youngest England player to feature at a major tournament when he started the 1-1 draw with France in Donetsk.
He was part of Hodgson's 23-man World Cup squad in 2014, but did not feature after injuring a knee during England's final warm-up game against Ecuador.
England will face Russia, Slovakia and Wales in Group B at Euro 2016, which begins on 10 June.
England winger Alex Oxlade-Chamberlain has been ruled out of this summer's European Championship after suffering his second knee injury of the season.