Guy Verhofstadt, the European Parliament's Brexit co-ordinator, described the General Election result as "yet another own goal" for the UK.
Theresa May is understood to have no intention of resigning as Prime Minister after a disastrous night for her party.
"Our shared responsibility and urgent task now is to conduct the negotiations on the UK's withdrawal from the European Union in the best possible spirit, securing the least disruptive outcome for our citizens, businesses and countries after March 2019", Tusk said in his congratulation letter to May.
Talk in Britain that a different ruling coalition could seek a "softer" Brexit than May has proposed, possibly seeking to remain in the single market, is also problematic for the EU.
And if it takes a week or more to cobble together a viable government, then the talks may have to be delayed from their scheduled start on June 19, throwing the whole Brexit timetable into doubt.
Even if the prime minister survives, she will be more vulnerable to pressure from what David Cameron - her predecessor - called the "extremes" in her party, including from some Eurosceptics who would prefer Mrs May to walk away without a deal at all.
"I am happy @Jeremy Corbyn has managed to remarkably boost the Labour Party's position".
But European politicians such as Angela Merkel may feel emboldened by Mrs May's failure to secure a landslide election win. "Let's put our minds together on striking a deal".
For the most part, though, at this early stage European Union diplomats see the result as shifting the odds more towards a softer Brexit, rather than increasing the likelihood of a unmanaged, hard exit without a withdrawal agreement.
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And Lord Newby, the Lib Dems' leader in the Lords has told Sky News that those who disagree with the Tories' Brexit approach "will be emboldened". Which will be about whether they want a hard or a soft Brexit."Mr Brok, 71, added: "It's a bad result for the UK".
He went on: "We face some appalling difficulties in my opinion, this is a critical stage for us - politics is changing".
Giving Northern Ireland's leading Unionist party a deciding say on government policy could tie the hands of Britain's negotiator on sensitive issues over the border, as well as further raising political tensions with Republicans.
Brussels' chief negotiator Michel Barnier has said the talks would begin when Britain was ready, suggesting he would consider a short delay. "Whether the other side can even begin will have to become clear in the next hours and days, since if there's no government-no negotiations".
In March, May bound Britain to leaving the European Union, with or without a deal to settle legal uncertainties for people and businesses, on March 30, 2019.
Asked before the election if the talks could be pushed back, a European Commission source said: "They have got to agree the date, so yes".
Earlier, Mr Tusk warned Britain might end up with no deal on its withdrawal arrangements or future trade if it did not open negotiations in time.
"I hope that the British will be able to form as soon as possible a stable government".
And a shortened timetable for talks may make it easier for Brussels to force the United Kingdom to pay an exit bill which could add up to as much as £85billion.
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