Billy Greening, a Conservative councillor for Horsham District Council in Sussex, is said to have confronted Ash Sarkar, senior editor of left-wing Novara Media, Huck Magazine editor Mike Segalov, and prominent Labour activist Owen Jones at The Falcon pub in Wandsworth.
The group say they had just been canvassing when they were approached by Mr Greening, allegedly poking Mr Jones in the chest (a claim he denies).
He is said to have shouted "For the many, not the Jew", a play on the Labour Party's slogan under Jeremy Corbyn, later claiming he wanted to provoke a "discussion" of the anti-Semitism row which has engulfed the party in recent months.
Apologising on Twitter, he said: "I approached Owen and said something I did not think through. What I said was wrong, plain and simple.
"I realise what I said was wrong - there is no other way to put it...What I said (I think) was meant to provoke a reaction which (I hoped) would have led to a discussion about the situation in the Labour Party.
"I can now plainly see that was a ludicrous assumption and attempt to draw a well known commentator into a debate.
"I would like to extend an apology to you all in person, I know this something you might not accept, but offering an olive branch is the least I can do...
"Some people might say it's only taken me 'getting caught' for me to issue an apology. Hopefully in the long term this will be the turning point I needed, and being called out on saying stupid/wrong/horrible things will end."
Ms Sarkar highlighted a tweet 'liked' by Mr Greening in the past by a British nationalist account, reading: "Just so you lefties know, when you call me a racist or bigot or xenophobe, I actually get a little turned on."
She told the Standard: "I'm not sure what this guy was playing at and, for a second, I had to wonder if he really was a Tory councillor - who could be this stupid?
"What's worrying is that he seems to find this kind of humour funny. In the Labour Party, we have clear guidance as to what action is taken with incidents like this.
"I would like to know how the Conservatives are going to deal with this."
Mr Segalov, a Jewish Labour activist who has been working to erase anti-Semitism from the party, said: "Reading his apology, I get the impression he doesn't really understand what it is he's done wrong.
"When he was confronted about his comment, he immediately tried to laugh it off by saying it was 'just a joke' which, as an elected politician, really isn't good enough.
"Clearly he has a history of treating racism as a joke or, at the least, trivialising it. I would like to see disciplinary action taken because this really isn't an issue to be made light of for a cheap shot."
Speaking to the Standard, Owen Jones said: "His behaviour was totally outrageous and I think the fact that a Conservative councillor could treat anti-Semitism as some sort of partisan joke is disgusting.
"Anti-Semitism is a cancer and we are doing everything we can in the Labour Party to stamp it out.
"But that is made all the more difficult when we have people on the other side treating it as if it is some sort of joke."
The Standard approached Horsham District Council and Mr Greening for comment.