Ministers have been urged to stand up for civil servants following criticism of the UK's outgoing EU ambassador.
Dave Penman, leader of the FDA union for senior Whitehall staff, said there had been a "deafening silence" from senior government figures since the resignation of Sir Ivan Rogers.
In a note to staff, Sir Ivan attacked "muddled thinking" about Brexit.
Some politicians welcomed his resignation and said someone "committed to Brexit" should be appointed.
The European Commission said it regretted Sir Ivan's departure, describing him as "a very professional, very knowledgeable, while not always easy interlocutor and diplomat who always loyally defended the interests of his government".
Labour has demanded that the government give a statement on the resignation when Parliament reconvenes next week, following its Christmas and New Year break.
But former cabinet minister Iain Duncan Smith said Sir Ivan - whose warning that Brexit talks could take a decade was revealed by the BBC last month - was "not irreplaceable". He said that when a civil servant "starts going public", ministers "can no longer trust that individual".
Another pro-Brexit Conservative, Peter Lilley, said half of Sir Ivan's parting note to his fellow UK diplomats in Brussels read "like a CV" and the other half "like some degree of sour grapes".
"One has a feeling that many diplomats, Eurocrats, are actually in the business of trying to negotiate our way back in rather than committing to getting us out, and if Sir Ivan Rogers was in the former group, it's good he's gone," Mr Lilley told BBC Radio 4's The World at One.
By John Pienaar, BBC deputy political editor
For one minister - an enthusiast for Brexit - it was very simple: "You're either on board, or you're not. He wasn't. We move on." The minister sounded rather cheerful.
So, Sir Ivan Rogers had gone because his face didn't fit. Now the way was clear for a true believer in the opportunities opened up by the vote to leave the EU.
If only winning a good deal for Britain in its divorce from the European Union, and eventually on the terms of trade for the UK outside the EU, was half so simple.
Deep faith in the benefits of leaving the EU may, or may not, be an asset to whoever ends up sitting behind the Union Flag when talks finally begin after March. No-one can claim it will be a sufficient qualification, and to be fair, no-one does.
Former UKIP leader Nigel Farage said a "tough Brexiteer" should replace Sir Ivan, while Tory MP Dominic Raab claimed Sir Ivan's "heart hasn't really been in Brexit".
But former Foreign Office permanent secretary Lord Ricketts said he was "really concerned at the undertone of denigration" of Sir Ivan and said it was vital for ministers to "stand behind" his successor.
And Mr Penman said he had "a lot of concern" about some of the comments, adding: "What surprises me is the deafening silence from ministers who should be taking to the airwaves to defend the integrity and capability of the impartial civil service."
He added: "If the civil service is to deliver a successful Brexit negotiation, the recipe for that success is unlikely to be to starve it of resources, lack clarity of objective and be surrounded with yes men and women who will not speak truth unto power."
The government said Sir Ivan, who had been due to leave his post in October, was quitting early so a replacement could be appointed by the start of Brexit talks, which Theresa May has promised to trigger by the end of March.
In his farewell note to fellow UK diplomats in Brussels, Sir Ivan said: "I hope you will continue to challenge ill-founded arguments and muddled thinking and that you will never be afraid to speak the truth to those in power.
"I hope that you will support each other in those difficult moments where you have to deliver messages that are disagreeable to those who need to hear them."
He said the government would only succeed if it "negotiates resolutely", adding, in a reference to the remaining 27 EU states: "Senior ministers, who will decide on our positions, issue by issue, also need from you detailed, unvarnished - even where this is uncomfortable - and nuanced understanding of the views, interests and incentives of the other 27."
Downing Street said it would not comment on the resignation email, although Sir Ivan was "free to express his own opinions".
By James Landale, BBC diplomatic correspondent
Outgoing ambassadors often write valedictory notes but few are as explosive as this. Sir Ivan does not name names but his remarks do not need much decoding.
The clear implication is that the advice the UK's ambassador to the EU has been giving has fallen on deaf ears in Downing Street.
And this is why he is standing down, depriving the government of one of its most experienced Brussels hands just weeks before negotiations over Brexit are due to begin.
His resignation took Downing Street by surprise, coming some eight months before his job was due to end or be extended.
So the hunt is now on for an envoy who can speak truth to power while retaining the confidence of Downing Street, who knows Brussels but is not seen as being part of Brussels.
Sir Ivan is expected to stay on for a few weeks while such a diplomatic paragon is found.
A government spokeswoman said: "Sir Ivan Rogers has resigned a few months early as UK permanent representative to the European Union.
"Sir Ivan has taken this decision now to enable a successor to be appointed before the UK invokes Article 50 by the end of March. We are grateful for his work and commitment over the last three years."
For Labour, shadow Brexit secretary Sir Keir Starmer urged Brexit Secretary David Davis to come to the House of Commons next Monday to answer MPs' questions on Sir Ivan's resignation.
He called it "frankly astonishing" and "deeply concerning" that he had written in his email that he did not yet know the government's objectives for Brexit.
Sir Keir added: "Time is running out. It is now vital that the government demonstrates not only that it has a plan but also that it has a clear timetable for publication."
Earlier Mr Duncan Smith said Sir Ivan "knew very well what he was doing" when he sent the note.
He added: "You must have absolute trust and cooperation. You cannot have this stuff coming out publicly."
Shadow international trade secretary Barry Gardiner said government policy "appears to be to shout down anyone who tells them things they don't want to hear".