That's the short answer. And when it comes to at-will employment, you don't need much more. At-will employment means either side can terminate employment, for any reason or for no reason at all.
Still, there are some limitations to at-will firings. And after any high-profile firing based on an employee's statements, many will wonder if the First Amendment or free speech principles apply. So let's take a look.
Free Speech ≠ Consequence-Free Speech
The plain text of the First Amendment reads: "Congress shall make no law ... abridging the freedom of speech." (Emphasis added.) That means that the government can't limit free speech. There are some exceptions to that rule, of course -- threatening speech is illegal -- just as there may be some statutory limits on what you can fire an employee for saying.
But no court has determined that firing an employee for bigoted speech is illegal. And asserting, in a 10-page 10th-grade-level manifesto, that the "distribution of preferences and abilities of men and women differ in part due to biological causes and ... these differences may explain why we don't see equal representation of women in tech and leadership" is just that. In fact, given the allegations of sexual harassment and discrimination in tech companies (the U.S. Department of Labor is currently investigating whether Google has unlawfully paid women less than men), not firing the memo's author may have opened the company up for litigation.
So while the government, in most cases, may not arrest you for your speech, your private employer, in most cases, can fire you for it.
Policies, Contracts, and Limitations on Termination
Some companies, while ostensibly free to fire their employees for whatever reason, may still choose to limit that power, whether through internal termination policies or employment contracts. An employment contract may include a clause requiring the employer to show good cause before terminating the employee. And employers can get in trouble for not following their own termination policies.
Neither seem to be in play in this case. The anti-diversity memo's author, James Damore, doesn't appear to have an employment contract with Google. And even if he does, Google's Chief Executive Sundar Pichai told employees that portions of the anti-diversity memo "violate our Code of Conduct and cross the line by advancing harmful gender stereotypes in our workplace," according to Reuters.
So, even though Google could've fired Damore for no reason, it had a good reason to do so.