There's no right to free speech in the work place. If he was fired for something he wrote on his private blog, or discussed with coworkers at an off-site bar, that might be different.
If he does sue, Google will have little problem showing that the employee, through his memo, has created a hostile workplace for women at Google.
For example, in California it is illegal to fire someone for "political activity", which this memo, debateably, may have been.
It is also illegal in California to fire someone for bringing up workplace concerns (this falls under union/worker organizing laws).
Finally, and most ironically, it is illegal to fire someone for bringing up discrimination concerns, EVEN IF those concerns turn out to be false/not illegal/not discrimination.
This guy, debateably, was bringing up discrimination concerns.
How it could be more political or about discrimination concerns?
"6 Instead set Googlegeist OKRs, potentially for certain demographics. We can increase representation at an org level by either making it a better environment for certain groups (which would be seen in survey scores) or discriminating based on a protected status (which is illegal and I’ve seen it done). Increased representation OKRs can incentivize the latter and create zero-sum struggles between orgs"
....discriminating based on a protected status (which is illegal and I’ve seen it done)....
Wonder if he has proof?
Can you point out the part of his memo that created a hostile environment for women?