A win over Motherwell was not enough and Foran, who wants to stay on as manager, felt let down by some players.
The manager made reference to "two or three bad apples in the dressing room".
And he added: "I probably should have got rid of them in January. I stayed loyal to a lot of players and some of them didn't pay me back."
Inverness won three of their last four Premiership matches as they fought to finish in the play-off spot.
But Hamilton Accies' 4-0 hammering of Dundee on Saturday ensured they will face Dundee United for a spot in the Premiership next term, while Inverness will play Championship football despite their 3-2 win over Well.
"It's obviously disappointment," Foran added. "The best clubs in the world get knocked down, it's how you rebuild and come back. I've told the players there are far worse off people in life.
"But it happens in football. It is not all rosy. It is not all about winning all the time. I've been part of relegation teams - you stay loyal, you rebuild. You get knocked down, you get back up again.
"You need to get rid of the two or three players you don't want, that haven't given 100%, and I look forward to getting rid of those three.
"Of the starting 11 today, I would hope to have at least 10 of those for next season [midfielder Greg Tansey is leaving to join Aberdeen]. I am very proud of the players and when we got the right team playing, we could finish well, with guys giving their all for the club."
Foran, who was a player at the time, stayed with Inverness when they were relegated from the top flight in 2009.
One year into a four-year contract, he hopes to do the same as manager.
"I want to be here next season," he said. "I will have a chat with the board probably on Monday or Tuesday and will find out what their thoughts are. But I am 100% behind this club and I expect to be here next season.
"There are changes to be made, on and off the field. Personally I want a smaller squad.
"I have got to look at myself as well. I haven't performed well enough. I have learned a heck of a lot this season - about myself, about players, the trust of people around you, and people you don't trust.
"One of my favourite seasons as a player was getting promoted, because we stuck together. I was part of the relegated team and helped them come straight back up, and I hope to do the same next season."
One player who will not be there is Tansey, who signed a pre-contract agreement with Aberdeen in March.
The midfielder, 28, was dismayed to leave on the back of relegation but felt he had given his all for the cause.
"It is never nice, but I can look myself in the mirror," said Tansey, who scored his seventh goal of the season in the win over Well, having scored nine last term.
"I gave all I could and have finished very similar to last season goals-wise. Sometimes things happen like that. You just have to move on as a professional footballer and get on with it.
"We probably thought we were too good to go down, but we have got to be at it every game."
Manager Richie Foran says Inverness Caley Thistle must rid themselves of "two or three bad apples" after they were relegated from the Premiership.