Taoiseach Leo Varadkar has said the threat level against politicians has risen in recent times and that what happened outside Leinster House was wrong.
TDs and staff were unable to leave the grounds of the national parliament for several hours on Wednesday afternoon due to the incident.
Ceann Comhairle Seán Ó Fearghaíl said that a review of security at Leinster House, involving gardaí and senior Oireachtas officials is under way and the matter will be "carefully considered".
He was speaking after a meeting involving an assistant garda commissioner and two senior officers, Clerk of the Dáil Peter Finnegan, Oireachtas Superintendent Barry Ryan and himself.
Mr Ó Fearghaíl said the long-standing right to demonstrate is essential and must be protected.
The Ceann Comhairle said that security at the Oireachtas was under "active" consideration and there would be "ongoing engagement" over the coming days, after which he anticipated there would be a "set of proposals" on how to deal with similar situations.
He said there was "every expectation" that the events could be repeated in the future, adding that gardaí had outlined their concerns at the meeting and the Oireachtas officials had shared theirs.
It comes after the Taoiseach said the right to a peaceful protest was essential in a democracy, but what happened was not peaceful.
"What I saw happening outside Leinster House was not a peaceful protest. There was violence, there was intimidation and that was wrong," according to Leo Varadkar.
He said a rising trend in violent threats against politicians had led to Garda Commissioner Drew Harris restoring garda protection to ministers recently.
Speaking in Miami, where he opened a new Irish Consulate General, he said that while there had always been threats against politicians, increasingly those threats came from people with convictions.
"Often they come from people who make idle threats or threats that are not credible, but increasingly the briefing from the commissioner is that threats are being made against politicians and public figures by people who have histories of violence and have convictions. So that's where it becomes more serious."
Mr Varadkar said that he did not fear anything like the 6 January attack on the US Capitol taking place in Ireland. "I don't think it was on that scale," he added.
He said he appreciated the level of protection afforded to politicians by gardaí and did not agree with suggestions that a more robust style of policing was required.
"There's a way to police protests and I know some people would like to see a more heavy-handed approach from the gardaí. That's not always the right approach.
"Though a lot of people in fringe groups, they're agents provocateurs. They almost want there to be an excessive reaction. And I think the gardaí have to judge that and they judge it very well."
Tánaiste Micheál Martin said the behaviour of some of those involved in the demonstrations was almost "fascist like."
Speaking at the National Ploughing Championships in Co Laois, Mr Martin said he believed those involved were "trying to create an alternative reality" aimed at undermining Irish democracy.
"Not only did they attempt to undermine the seat of Irish democracy in terms of the Parliament, the Dáil and the Seanad, they also upset and undermined people going about their daily business in Dublin, who are not at all impressed with the nature of the protests, if you could call them that.
"I think some of these people are trying to create an alternative reality in terms of the country, undermining our democracy and some of it almost approximates to sort of fascist like behaviour in terms of intolerance.
"In terms of not allowing people to go in and out of a democratically elected parliament. It's a very serious issue."
Mr Martin said public representatives must be allowed go about their work and represent the people who elected them.
He added freedom of speech is very important in the parliament and populism should be avoided by all public representatives.
'Disgraceful' scenes have 'no place in our democracy' - McEntee
Minister for Justice Helen McEntee said the "disgraceful" scenes which took place outside the Dáil yesterday had "no place in our democracy".
Ms McEntee said that the Government will "always protect the right to peaceful protest", but that scenes yesterday were the "intimidation and threatening of elected representatives and members of An Garda Síochána doing their work on behalf of the people and the State".
She said she has spoken to Garda Commissioner Drew Harris about the incident and told RTÉ's News at One added that those involved were "a small minority of people".
"They do not represent the majority of people, and it cannot be the situation that something like this is allowed to happen or that it is allowed to continue in this way," she said.
She said she does not want "a situation where we cannot walk about or down the street for fear of somebody intimidating or harassing us" and that she would not like to see access to Government Buildings restricted.
"We see it in many other countries where you cannot get near your parliament, where you have no access at all," Ms McEntee said. "I would not like to see that happen, and I would not like to see the actions of a few small people, a minority to change that."
A 'fundamental attack on democracy' - Ceann Comhairle
Ceann Comhairle Seán Ó Fearghaíl earlier condemned the "fundamental attack on democracy", adding: "We can never see again happen what happened outside yesterday."
He said if the situation went unchallenged "it will bring us to a point that we saw the United States in the aftermath of the last presidential election".
"1,200 people work here, 220 of us politicians. The people that suffered abuse yesterday came from all walks of life. Civil servants, political staff, political advisors, journalists, the whole spectrum, and what happened was particularly vile and vicious.
"The people who are involved in that, they offer nothing of any benefit to any cause."
Mr Ó Fearghaíl also told the Dáil that after the summer recess was called he "had a lengthy meeting with Commissioner Harris to draw his attention to the pressure that members are under, and the concern that members have for their staff and for their families, none of whom have signed up to take the abuse".
The "appalling scenes" outside Leinster House yesterday went beyond "legitimate protest", and that the people protesting were going against "representative democracy", Minister for Public Expenditure Paschal Donohoe said.
There is an "urgent" need to look at security but reviews have taken place in Leinster House, which prompted changes to access and entry points, he said.
FG senator sends letter to Harris
After politicians needed gardaí help to get into their place of work, Fine Gael Senator Barry Ward has written to Commissioner Harris.
He asked what steps would be taken to protect the country's democratic institutions.
Speaking on RTÉ's Today with Claire Byrne, Senator Ward said scenes like this cannot happen again and a "marker needs to be set down" around this "thuggery".
"If there is a feeling that they can get away with it, that they can behave with impunity then it will happen more, it will escalate, people will be injured and worse," he said.
"I would hate to see a small group of people are allowed to intimidate us into reducing access to somewhere like Leinster House which belongs to everyone."
Mr Ward said that since the foundation of the State, TDs and senators have had a right to attend parliament but that right was disrupted.
Gardaí need to be well equipped to deal with people who are not acting in a reasonable way, he added.
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13 held over public order incidents around Govt Buildings
Cathaoirleach of the Seanad Jerry Buttimer called for a policing plan to guarantee politicians access to both sides of Leinster House.
"No member or staff should be prevented from leaving their work or subjected to vitriolic behaviour," he said.
Mr Buttimer added that staff at Leinster House feared injury or harm and were very upset.
"The perimeter at Kildare Street and Merrion Street must be a sterilised zone and it's important now that the Oireachtas, gardaí and city council sit down and prepare a policing plan," he said.
Speaking on RTÉ's Morning Ireland, Mr Buttimer said: "I have never witnessed the behaviour and the thuggery of yesterday."
The proactivity of gardaí ensured no one was seriously injured during the incident, he said.
In a statement, gardaí said that a policing operation was put in place to facilitate the resumption of the 33rd Dáil.
"On a number of occasions gardaí were required to intervene during public order incidents which occurred on Molesworth Street, Kildare Street and Merrion Street," it said.
"Regular uniform gardaí were supported by plain clothes units and the Public Order Unit in 'soft cap' mode.
"In total 13 arrests were made over the course of today connected with incidents in the vicinity of Government Buildings," the statement added.
Tánaiste Micheál Martin described the disturbing imagery of a gallows held at the demonstration as "reprehensible".
Additional reporting Mícheál Lehane, Sean Whelan
