Plaid Cymru leader Leanne Wood has said she is "staggered" by the response to her suggestion that the Barcelona attack was "far right" terrorism.
Thirteen people died when a van drove into pedestrians in the Spanish city, with so-called Islamic State (IS) saying it was responsible.
The Welsh Conservatives called on Ms Wood to apologise or resign, while UKIP accused her of an "outrageous smear".
Ms Wood said far-right ideologies drove both IS and white supremacists.
People from 34 different countries were injured or killed in the attack on the famous Las Ramblas on Thursday.
Hours later, police killed five suspected jihadists in a second vehicle attack in the town of Cambrils. A woman injured in that attack died later, taking the total death toll to 14.
Following the Las Ramblas attack, the Plaid Cymru leader posted her reaction on Twitter, saying: "Ofnadwy / terrible. Is this more far right terrorism? My thoughts are with all those affected."
She added: "All forms of political violence are the same. USA, Barcelona, everywhere. They are ideology-driven & we have to understand that to stop it."
The comments sparked a critical response from politicians of other parties, and on social media.
Welsh Conservative leader Andrew RT Davies, who had tweeted it was "sickening" to hear of the attack in Barcelona, said it was "unbelievable" Ms Wood had speculated it might be linked to a far-right group.
His fellow Tory assembly member Janet Finch-Saunders said the comments were "at best poorly timed and ill-judged, at worst cynical and morally depraved".
"Politicising terror so fresh after an attack can never be acceptable, and only adds to the hysteria and toxicity of debate.
"As leader of a so-called progressive party, she should show some contrition by issuing an immediate apology and, if she doesn't, I think she should resign."
A tweet from UKIP Wales said: "Absolutely outrageous smear against UKIP which you have called 'far-right' countless times. How dare you associate us with murderers!"
Later, UKIP's Senedd group leader Neil Hamilton said it was a "gross insult" to more than 130,000 people who voted for the party in the 2016 assembly election.
"Leanne Wood is clearly not fit for purpose," he said.
"She is an embarrassment not only to Plaid Cymru but to the whole of Wales."
'No better than each other'
Following the outcry, the Plaid Cymru leader responded on Facebook, saying: "I am staggered by the reaction to the point that Isis and white supremacism both have far right ideologies driving them.
"Both see their group as superior to others. Both see people who are not in their group fair targets for abuse, violence and even death. Both hate minorities and consider women to be less than men.
"Both believe in using extreme violence to repress people with different views. This is far right/ fascism ideology. How can it not be?"
Ms Wood said she could see how her comments would "upset" people on the far right, saying "the comparison links western racists and white supremacists to their number one hate target and shows them as no better than each other".
She ended her post by saying: "While it's important, in my view, to say and keep saying that politics and ideology lie behind these acts of violence, my thoughts are first and foremost with the victims of last night's horror."
Ms Wood also defended her comments in a statement issued by Plaid Cymru later on Friday, describing IS as a "fascist organisation".
"It is driven by an ideology of hatred and violence that scapegoats minorities and considers women to be worth less than men," she said.
"This far-right ideology and the violence that goes with it must be rejected and opposed wherever it rears its head - be it Islamic fundamentalism or white supremacism."
Meanwhile, First Minister Carwyn Jones said: "Our thoughts today are with all of those who were affected by yesterday's terrorist attacks on Barcelona and Cambrils.
"It is truly horrifying to think that people who were simply going about their daily lives, relaxing and enjoying themselves could be targeted in such an evil way."
Mr Jones added Barcelona was "a proud and beautiful city", and that Wales shared "many historical and cultural ties" with the Catalunya region of Spain.
"I want to end by expressing my deepest sympathies on behalf of the people of Wales," the first minister said.
"We stand by your side at this difficult time."