Despite a win against Wales, draws with Russia and Slovakia meant the Three Lions had to settle for second place in Group B behind Chris Coleman's side.
That means England will now play Iceland in the last 16 next Monday, with hosts France potential quarter-final opponents if they can defeat the Republic of Ireland on Sunday.
"There are certainly teams that have impressed me, but I do think it is wide open," vice-captain Cahill told reporters.
"You see the French, Spanish and Germans who are always there, but is there one team that is going to run away with it? I don't think there is.
"We have to go again and I think show the same mentality that we have shown in the first three games.
"You see how hard teams are having to work, like the French in the second game [against Albania]. It was 2-0 and seemed like an easy game, but it was far from an easy game going into the last five minutes.”
He added: "You want a team to come out and play against you a bit more. Everyone is fighting to the death so it's very tough. But we should be pleased with what we are doing, no two ways about it.
"Apart from the obvious, putting the ball in the back of the net, everything you look for is there: commitment, playing with confidence, creating chances.
"I'm not making excuses because we have had the majority of chances and the stats show that we should be winning the games.
"The positives are all there. We are going out there on the front foot and taking the game to them. I have never been involved in such a one-sided game as that second half against Slovakia and in the next game we should go out there and do something similar."