In a statement he said Mr Woolfe had “subsequently collapsed” and his “condition is serious”.
Mr Woolfe, MEP for the North West, was taken to hospital in Strasbourg for tests.
Mr Woolfe announced on Wednesday he will stand for the party’s leadership after Diane James stepped down.
In his statement, Mr Farage said: “I deeply regret that following an altercation that took place at a meeting of UKIP MEPs this morning that Steven Woolfe subsequently collapsed and was taken to hospital. His condition is serious.”
UKIP MEP Roger Helmer told the BBC the incident followed “some lively exchanges” at a closed meeting of party MEPs on Thursday morning. He said the gathering happened “a good two hours” before a voting session from which he understood Mr Woolfe left.
Raheem Kassam, the other candidate to have declared for the UKIP leadership so far, tweeted his best wishes for Mr Woolfe, as did ex deputy chairwoman Suzanne Evans.
Mr Kassam tweeted that he had cancelled his appearance on BBC2’s Daily Politics “out of respect” for Mr Woolfe, following reports he had been taken ill.
He added: “I really hope @Steven-Woolfe is okay. Plz send him your best wishes.”
Ms Evans, who is said to be considering her own leadership bid, tweeted: “Shocked to hear Steven Woolfe has apparently collapsed in the European Parliament in Strasbourg. Wishing him well for a speedy recovery.”
Diane James said on Twitter: “My thoughts are with @Steven-Woolfe and his wife and daughter at this anxious time and I wish Steven a speedy and full recovery.”
Bill Etheridge, a former UKIP leadership contender and an MEP for the West Midlands and Dudley, tweeted: “Hoping all is well and wishing @Steven_Woolfe all the best after he was taken ill today in Strasbourg.”
Mr Farage said on Wednesday he would return as UKIP’s interim leader until a fresh election could be held to find Ms James’s successor.
Ms James announced her resignation after just 18 days in the job, citing professional and personal reasons for her decision.
She succeeded Mr Farage on 16 September after he resigned following the Brexit vote.
Mr Woolfe was unable to take part in the previous UKIP leadership race after he missed the deadline for submitting his nomination.