Mr Abbott says he was assaulted by a same-sex marriage supporter on Thursday afternoon while walking to his waterside hotel after a meeting at the office of The Mercury newspaper.
Tasmanian police on Friday appealed for the man, who is believed to be aged in his 40s, to turn himself in.
They say he has short spiky sandy strawberry coloured hair and facial piercings and was wearing a leather jacket with a ‘yes’ badge at the time.
Investigators are working through CCTV footage from the area and have spoken to a number of witnesses.
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But Mr Cerritelli wouldn’t confirm whether Mr Abbott and his alleged attacker exchanged words outside Customs House Hotel.
“We will not get involved in any political matters of yes or no,” he said.
Police say the former prime minister suffered minor injuries but didn’t require medical assistance.
Mr Abbott made a formal complaint on Thursday night after being contacted by police but was “quite calm” when talking to investigators.
Mr Cerritelli was confident the matter will come to a quick resolution and has urged members of the public with information to come forward.
0:00 Tony Abbott speaks about the moment he was ‘headbutted’ by man wearing a same-sex marriage badge Share Tony Abbott speaks about the moment he was ‘headbutted’ by man wearing a same-sex marriage badge
Mr Abbott commented on the incident on Friday morning
“He said, ‘I want to shake your hand’. I saw him coming towards me,” he told reporters.
“I held out my hand, he grabbed my hand and it turned into a head butt. Now, it wasn’t a very effective head-butt, obviously, but it certainly was a head-butt. I pushed him away.
“A staff member of mine grappled with him briefly and he scar perked off saying F this, F that ‘You deserve it because of what you’ve said'”
Prime Minister Malcolm Turnbull has condemned the alleged headbutt attack Mr Abbott, labelling it “disgraceful”.
The prime minister rang his predecessor after the alleged incident in Hobart on Thursday evening and was in in touch, via text, again on Friday.
0:00 Prime Minister Turnbull speaks to 3AW radio about Abbott ‘headbutt’ Share Prime Minister Turnbull speaks to 3AW radio about Abbott ‘headbutt’
“This is a disgraceful incident and I condemn this assault on Tony,” Mr Turnbull told Neil Mitchell on 3AW radio.
Mr Turnbull, who also spoke to the Australian Federal Police commissioner, hopes the assailant will be identified and charged.
“It’s just a reminder of how ugly this debate is getting,” Mr Abbott said.
Mr Abbott was in Tasmania campaigning for the ‘no’ vote in the national same-sex marriage survey along with Liberal colleague Eric Abetz.
Senator Abetz had dinner with Mr Abbott after the incident and described the former prime minister as being in “fine form”.
“He was stirred, but not shaken,” the senator told Sky News.
He agreed the incident was out of character for the broader ‘yes’ campaign but said ugliness on social media showed some would be empowered if the vote was successful.
“This is just a bit of a harbinger of what is likely to occur, and it’s not the sort of Australia I want,” Senator Abetz told ABC TV.
Post by Tony Abbott.
Cabinet minister Christopher Pyne said no one should be attacked for having a different view on marriage and it must have come as a nasty shock for Mr Abbott.
“It is an un-Australian thing to do and I hope that Tony is okay,” he told the Nine Network.
Deputy Liberal leader Julie Bishop echoed that sentiment.
“Violence of any form is never acceptable,” she told reporters in New York.
Opposition Leader Bill Shorten described the attack as unacceptable.
“I’m glad Mr Abbott isn’t seriously injured and I’ve rung him to say so,” he said on Twitter.
Great to have @TonyAbbottMHR in Tas over last few days. Disappointed a yes campaigner would try to assault fmr PM. Must be condemned by all.
— Eric Abetz (@SenatorAbetz) September 21, 2017There is absolutely no place for violence in the marriage equality debate. This is about treating people fairly and with respect & dignity.
— Alex Greenwich MP (@AlexGreenwich) September 21, 2017This is the statement of respect that all staff and volunteers of The Equality Campaign must abide by. pic南京夜生活,/liYUVBknYI
— AU Marriage Equality (@AMEquality) September 21, 2017
0:00 Same-sex marriage around the world Share Same-sex marriage around the world