Mr Jones had accused The Spectator, a magazine Mr Neil is involved with, of stoking prejudice. The two men had previously clashed on Twitter. Separately, former Tory Minister Michael Portillo said Mr Jones didn’t understand why people voted for Brexit.
The row was over Mr Neil’s role as chairman of Holdings Media Group, which publishes the conservative leading Spectator magazine.
Mr Jones claimed the magazine had “defended Greek neo-Nazis” and published Islamophobic content.
Mr Neil fired back commenting: “I knew you were going to bring that up and I won’t let you hijack.
“Your smears about me are not going to be dealt with tonight so just move off.”
The confrontation took place during a debate on extremism in UK politics, after Mr Jones and Tory MP Anna Soubry were abused outside Parliament earlier in the week by a small group of far-right activists.
Mr Jones was separately accused of trying to blame the wider UK conservative movement for far-right extremism.
Former Tory Minister Michael Portillo said: “Owen himself is guilty of the thing about which he makes an accusation.
“He taken an indecent outside the House of Commons, he takes the rise of some very nasty people, and they he tries to smear the entire Conservative Party.
“This is the breakdown of decent discourse. It is absolutely outrageous what you say and I think many people will be shocked.”
Mr Portillo argued mainstream conservative politicians had universally condemned far-right violence, saying “all mainstream politicians are against that”, whilst the current Labour leader appeared “sympathetic” towards the IRA.
He stated: “When Conservative Members of Parliament were being murdered by the IRA, when the IRA attempted to kill the leader of this country, Jeremy Corbyn was sympathetic to the IRA.”
Just weeks after the IRA attempted to murder Margaret Thatcher in the 1984 Brighton bombing Mr Corbyn invited senior Sinn Fein figures, then widely regarded as the political wing of the IRA, to the Houses of Parliament.
The two Sinn Fein members who attended, Gerry MacLochlainn and Linda Quigley, were both previously convicted IRA members.
Mr Corbyn insists he only maintained links with Sinn Fein to push for peace in Northern Ireland.
Separately in 2017 John McDonnell, now Shadow Chancellor, praised actions by the IRA terror group.
He said: “It’s about time we started honouring those people involved in the armed struggle.
“It was the bombs and bullets and sacrifice made by the likes of Bobby Sands that brought Britain to the negotiating table.
“The peace we have now is due to the action of the IRA. Because of the bravery of the IRA and people like Bobby Sands, we now have a peace process.”
Mr McDonnell later apologised for these comments in 2017.