However, the party's education spokesperson stood by his views insisting that he continued to believe in the basis of his argument. He added that his apology "should not take away from the substance of what I said".
Mr Hayes accused "politically correct opportunists" of seizing a slip of the tongue to create an irrelevant argument.
Mr Hayes, who is in with a strong chance of taking over from Enda Kenny as the next leader of FG, had called for immigrant children with poor English to be "segregated" from other public schools.
Now he has said: "I should not have used the word 'segregation'. It was wrong and clumsy and it is my fault and I should have realised what the reaction would be.
"I should not have used the ridiculous word I did, and apologise to anyone who was offended by it."
Despite this, he continues to believe in the basis of his argument, adding: "It is not fair on the pupils coming into our schools -- some of whom do not have a word of English and cannot even write their names -- or the teachers who want to help them and other pupils with good English," he said.
The FG frontbencher hit out at Fianna Fail Minister of State Martin Mansergh in particular, for his comments that Mr Hayes' had evoked images of racial segregation.
"Opportunistic politicians should not jump on the bandwagon," said Mr Hayes. "Please debate the issue rather than the relevance or irrelevance of a word.
"We should be able to raise sensitive issues without being called a racist."
Mr Hayes caused controversy when he said that many parents were frustrated at the effect the lack of segregation was having on the education of their English-speaking children.
The Irish National Teachers' Organisation (INTO) said the proposal to separate students with poor language skills was "discriminatory, inequitable and deeply flawed".
The Association of Secondary Teachers Ireland (ASTI) said it supported the idea of separate classes initially for some immigrant children that would focus on English language skills.