Former Google engineer James Damore is being held up as a free-speech hero by his supporters, but his dismissal by the tech giant might may be more of an ethical morass than a legal issue.

Damore is the center of a controversy over his internal memo that claims biological factors contribute to inequality in the tech sector, such as his view that women have higher "neuroticism" than men. After the memo went public, Google (GOOG) fired Damore, with Google CEO Sundar Pichai denouncing it as "advancing harmful gender stereotypes."

Damore responded by claiming he had been "wrongfully terminated" and vowed to pursue legal action.

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While it's unclear what legal route Damore might pursue, free-speech grounds would be a stretch, given that the First Amendment provides protection only from government censorship.

Of course, employees enjoy legal protections at work, such as anti-discrimination laws for protected groups. But the First Amendment doesn't prohibit an employer from firing a worker based on their speech, said Lee Rowland, senior staff attorney at the American Civil Liberties Union's Speech, Privacy & Technology Project.

"As a private company, Google has the right to discipline employees for their speech or for violations of their code of conduct," Rowland said. "But every company should use that power judiciously to avoid creating a perceived political litmus test for employment."

In a statement to CBS News on Tuesday, Damore said he submitted a charge to the National Labor Relations Board about "how Google's upper management is misrepresenting and shaming me in order to silence my complaints. It's illegal to retaliate against a NLRB charge."

But simply filing an NLRB charge doesn't mean a company can't fire a worker for violating its standards or code of conduct.

"It is accurate to say that retaliating against an employee because he or she has filed a complaint is illegal," Rowland said. "However, filing a complaint is not an absolute barrier to other lawful discipline or termination."

What's clear, however, is that Google has waded into an ethical thicket.

"It's not a great idea for companies to overly control or censor their employees' speech, particularly outside of the workplace," Rowland noted. "A company will be attractive to a much smaller percentage of the workforce if they become known as an employer that micromanages their employees' beliefs and speech."

Corporations are increasingly wary of how their employees express themselves on social media and how they use time or resources while at work to sound off about their views. While a worker may feel it's appropriate to use a work account to tweet about gender bias, an employer might not agree, especially if that view makes co-workers or clients uncomfortable.

"The use of company resources to spread highly divisive political thought is inappropriate," said Aine Donovan, director of the Ethics Institute at Dartmouth College. "Whether the content was liberal or conservative, it's not appropriate for an employee to misuse resources in this way."

She added, "He was fired not merely because of his opinions, but because he misused company resources."

It may seem ironic to his supporters that Damore was fired after writing that he heard some co-workers say they were fearful of expressing the same views because they were worried about getting fired. In their view, Google's actions appear to support the "ideological echo chamber" that Damore complained about.

Yet his supporters may be overlooking what ethics and human-resources experts say is a major flaw with his manifesto: Rather than seeking out productive ways of discussing his views with the company's management, the memo "reads like the condemnation of an entire system," as Donovan noted.

"He surely must have been aware that a new HR director was charged with addressing some of the issues he touched upon," she said, referring to Google's recently hired head of diversity, Danielle Brown. "Improving an organization doesn't start with throwing a grenade into the conference room. Instead, it requires thoughtful consideration about making voices heard that represent a dissenting point of view."

Brown responded earlier with her own memo, describing change as "often uncomfortable" but stressing that Google believes "diversity and inclusion are critical to our success."

Still, Google risks alienating employees who may now feel they can't voice their honest opinions at work, said Autumn Manning, the CEO of human-resources tech platform YouEarnedIt. While she said she doesn't agree with Damore's view, she said he should have been able to express his views, although he should have used a more appropriate format for opening a discussion.

"The most effective cultures are the ones where people can bring their true selves to work," Manning noted.

Because gender and racial bias can lead to difficult discussions, companies may want to consider how to encourage workers to air their concerns or complaints, she said. That could mean setting up meetings where employers can discuss touchy subjects in a private forum or anonymous surveys.

Even so, sometimes employers and employees just aren't the right fit.

As Dartmouth's Donovan said, "The bottom line in business ethics is that sometimes there simply isn't a "match" for an employee and an employer. Take the high road and find an organization that better suits your philosophical/political beliefs."

A better fit for Damore might not be hard to find. On Tuesday morning, Wikileaks' Julian Assange wrote on Twitter that the organization was offering the ex-Googler a job. Noted Assange: "Censorship is for losers."