Koeman was furious after the 2-1 loss ended the Toffees' hopes of challenging for a trophy in his first season.
"I know what is needed to change," the 53-year-old Dutchman said.
"If everybody opened his eyes maybe we will get further in our improvement - because that is what this team needs."
Everton paraded new £11m forward Ademola Lookman, 19, before kick-off, but negotiations continue to drag on to secure Manchester United midfielder Morgan Schneiderlin.
It is not bad luck. We did it ourselves
Koeman wants major improvements to his squad and said: "I have that in mind and the board knows because we speak together about what we and the team needs and that is the job of [director of football] Steve Walsh.
"I hope everybody realises what the team needs to get results that are the expectation of this football club.
"I can explain more but I don't like to because that's really talks we need to do [internally]."
He added that the fact teenage Everton midfielder Tom Davies was one of the best players on show against Leicester "says enough" about the squad at his disposal.
Koeman also refused to put the cup exit down to ill fortune after seeing Romelu Lukaku's opener quickly cancelled out by two goals from Leicester substitute Ahmed Musa.
"It is not bad luck," he said. "We did it ourselves and it was just unbelievable how we dropped back.
"It was a lack of concentration at 1-0. I just can't see how they score the goal to make it 1-1. No pressing on the ball, a pass inside Seamus Coleman and Musa is faster - three defenders for us and it is 1-1.
"If you play like we did at 1-0 you ask for goals against. That is not the way you need to play at 1-0."
And with a Premier League match against Manchester City next up, Koeman warned his players they risk being on the end of a result similar to the 5-0 FA Cup drubbing Pep Guardiola's side dished out to West Ham on Friday.
"If we play like the last 30 minutes it can be the same result as against West Ham," he said.
Everton manager Ronald Koeman fiercely criticised his players and demanded action from the club's hierarchy in the transfer market after the FA Cup exit at home to Leicester City.