The Vatican joined Muslim figureheads from Indonesia and Egypt, as well as Switzerland, in denouncing the vote as a blow to religious freedom.
France's FM Bernard Kouchner expressed shock at the ban which, he said, showed "intolerance" and should be reversed.
But European right-wing groups welcomed the result, calling for other countries to take similar measures.
More than 57.5% of Swiss voters and 22 out of 26 cantons - or provinces - voted in favour of the ban on Sunday.
The proposal had been put forward by the Swiss People's Party (SVP), the largest party in parliament, which said minarets were a sign of Islamisation.
'Expression of intolerance'
The Vatican on Monday endorsed a statement by the conference of Swiss Bishops criticising the vote for heightening "the problems of cohabitation between religions and cultures".
Egypt's Grand Mufti Ali Gomaa described the ban as an insult to the feelings of the Muslim community in Switzerland and elsewhere.
Sunday's surprise result also prompted dismay from secular leaders in Europe.
"I am a bit shocked by this decision," Mr Kouchner told France's RTL radio on Monday. "It is an expression of intolerance and I detest intolerance.
"I hope the Swiss will reverse this decision quickly."
But Marine Le Pen, the deputy-leader of France's far-right National Front, praised the outcome, and said France should now hold a wider referendum on multiculturalism.
"The elites should stop denying the hopes and fears of European peoples who, without opposing religious freedom, reject ostentatious symbols forced on them by politico-religious Muslim groups, often verging on provocation," she was quoted as saying by Agence France Presse.
In Belgium, Italy and the Netherlands, anti-immigrant movements called on their own governments to debate similar measures.
"What can be done in Switzerland, can be done here," said Geert Wilders, the head of the Freedom Party in the Netherlands.
Meanwhile Roberto Calderoli, a member of Italy's Northern League, which is part of the country's ruling coalition, said: "Switzerland is sending us a clear signal: yes to bell towers, no to minarets."
In recent years, countries across Europe have been debating how best to integrate Muslim populations.
France focused on the headscarf, while in Germany there was controversy over plans to build one of Europe's largest mosques.
Government ignored
The vote is very bad news for the Swiss government which fears unrest among the Muslim community, says the BBC's Imogen Foulkes in Bern.
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The Swiss government had opposed the ban, saying it would harm Switzerland's image, particularly in the Muslim world.
Voters worried about rising immigration - and with it the rise of Islam - had ignored the government's advice, our correspondent adds.
The government said it accepted the decision, and that the construction of new minarets would no longer be permitted.
Justice Minister Eveline Widmer-Schlumpf said: "Concerns [about Islamic fundamentalism] have to be taken seriously.
"However, a ban on the construction of new minarets is not a feasible means of countering extremist tendencies."
She sought to reassure Swiss Muslims, saying the decision was "not a rejection of the Muslim community, religion or culture".
'Major problems'
Switzerland is home to some 400,000 Muslims and has just four minarets.
After Christianity, Islam is the most widespread religion in Switzerland, but it remains relatively hidden.
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There are unofficial Muslim prayer rooms, and planning applications for new minarets are almost always refused.
Supporters of a ban claimed that allowing minarets would represent the growth of an ideology and a legal system - Sharia law - which are incompatible with Swiss democracy.
But others say the referendum campaign incited hatred. On Thursday the Geneva mosque was vandalised for the third time during the campaign, according to local media.
Amnesty International said the vote violated freedom of religion and would probably be overturned by the Swiss supreme court or the European Court of Human Rights.
The president of Zurich's Association of Muslim Organisations, Tamir Hadjipolu, told the BBC: "This will cause major problems because during this campaign mosques were attacked, which we never experienced in 40 years in Switzerland.
"Islamaphobia has increased intensively."
Sunday's referendum was held after the SVP collected 100,000 signatures from voters within 18 months calling for a vote.
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SIC: BBC