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Several officials told CNN there is also fear of retaliation from an administration that they say has consistently sidelined America's oldest Cabinet agency.
"There is most certainly a sense within the building that the potential for reprisal is present," one former State Department official said.
The State Department has not responded to requests for comment regarding criticism of Pompeo.
Unfounded theories
Tensions within the department have been on clear display.
And Giuliani has pushed an unfounded theory that Ukraine masterminded 2016 presidential election interference. US intelligence has found no proof of that and has consistently said Russia was to blame.
Separately, the President has also had proxies reach out to foreign allies to investigate the origin of US intelligence probes into Russia's 2016 election meddling.
When asked Saturday whether the Ukraine investigations were appropriate, Pompeo -- a former CIA director -- raised no objection to the efforts to contradict US intelligence agencies and gave credence to Giuliani's unsubstantiated theories about Ukraine.
"Governments have an obligation, indeed a duty, to ensure elections happen with integrity, without interference from any government -- whether that's the Ukrainian government or any other -- so inquiries with respect are completely important," Pompeo said in Athens.
Pompeo also touched on the separate Trump effort to have allies investigate the origins of the US intelligence agencies' probe into Russian interference in the 2016 election. The Trump administration has reportedly asked the UK, Australia and others to help.
"If we need another government's assistance, it's very reasonable to ask that government," Pompeo said.
'Intimidate, bully & treat improperly'
The President raised the investigations in that call. White House officials subsequently tried to hide records of it on a classified computer server reserved for national security secrets.
"There are legitimate reasons why you might not want to send people up to brief" on Capitol Hill, the former State Department official told CNN. "Maybe they're not the best informed about something, maybe they're not very good in front of Congress, but that's not the case for an ambassador."
More than one official observed that Pompeo's robust declaration was doubly odd because he isn't generally seen as willing to defend the department, especially in the face of the President's policies or rhetoric.
There's a sense that "at the top level ... there's no real ability or willingness to stand up to what the President wants," a retired career diplomat said. "Even when the President has a bad idea, Pompeo is unable to stand up to that in the way [former Secretary of State] Rex Tillerson did."
The first former official said this perception feeds "a general feeling of fear that permeates the State Department. The current veteran career official says the result "is probably the lowest morale point that I have seen" in over two decades.
To learn Trump spoke that way to a foreign leader about a US diplomat was "chilling," said a career foreign service officer who recently left the department.
Trump has also issued seeming threats directly to department staff, telling them in late September at the UN that whistleblowers should be treated like spies who had committed treason and were then presumably hanged.
Deep unease
As the Ukraine saga unfolds, a third former career official who has spent decades in the department echoed others in describing a deep unease among staff about "having to do things that border on the political defense of the President, rather than executing American foreign policy consistent with their oath of office."
Despite the churn surrounding his agency, Pompeo kept to his schedule on a six-day trip to Europe that included a stop in his family's ancestral village in Italy. His aides refused to answer reporters' questions about the controversy or Pompeo's participation in the July 25 call between the leaders, which he did not immediately own up to. He did admit to being on the call on Wednesday.
One Pompeo ally said the Secretary is still fuming about Giuliani's claims that he reached out to Ukrainian President Zelensky's aides at the direction of the State Department. A source close to Pompeo strongly disputed that.
It's clear that Pompeo is trying to distance himself from Trump's lawyer -- unnamed Pompeo proxies have been appearing in media accounts saying he never promised Giuliani he would "investigate" allegations about Ukraine.
"We'll obviously do all the things we are required to do by law," Pompeo said during his Athens press conference on Saturday. And he dug in, repeating claims that there have been "congressional inquiries that have harassed and abused State Department employees by contacting them directly."
Congressional aides dispute that characterization.