Italy's 1-0 defeat to Costa Rica on Friday meant that England will not reach the knockout stages of a World Cup for the first time in 56 years.
The result means England will now only be playing for pride against Costa Rica on Tuesday following defeats to Italy and Uruguay in their opening two fixtures.
Manchester United striker Rooney said on social media that the England players were "devastated" to be dumped out of the World Cup at the first hurdle.
Rooney posted a message on his Facebook page that said: "Absolutely devastated to be out of the World Cup.
“Going into each game we had great belief in ourselves but unfortunately it hasn't worked out.
“Sorry to all the fans that travelled and at home that we haven't done better...gutted!”
England goalkeeper Joe Hart insists that the team remain firmly behind manager Roy Hodgson.
"Me personally, I speak for the players when I say we are proud to play for Roy Hodgson," said the Manchester City man.
"He is a great manager, a very passionate man, and someone I have an awful lot of respect for. I hope to continue playing for him."
Hart thanked the fans for their support and said they could rest assured that the squad had given it their all.
"I can't thank people enough for making this trip to Brazil, for making the efforts at home, for staying up for us, for revolving their work around us," he added.
“If I wasn't here myself, I would be doing the same. I can speak on behalf of the lads that we've not left anything to chance, we have tried everything.
"We haven't scored enough goals and let too many in. We are going to dissect it but we have committed our lives to this tournament, thrown everything we can at it.
"We have just not had the main stat, which is results."