The burden of proof for alleged discrimination is to stand up and say "I allege discrimination". Discrimination alleged. QED. You have it in writing, on the court transcript.
If Damore is clever enough to work at Google, he's probably clever enough to figure this out, so I predict 100% chance of success in this legal endeavour.
Unfortunately for Damore, the judge probably will probably award damages on the scale of one piping-hot mocha cappuccino (to be delivered upright, in a protective cup, with a spill-proof lid) and then assign the entirety of Google's legal costs to the plaintiff (Damore begins to faint), up to—but not exceeding—two hours of a discount public defendant, one H1-B dry-cleaning bill (it's just a second day job to pay the bills for an underfunded moonlight startup), and two cross-town Ubers (Damore perks up again like he just received a Mia Wallace special spiked with Adderall).
Google, with a driven corporate culture of work to completion, immediately delivers the requisite coffee to Damore, who expresses no surprise. (The inventor of Google Glass cut his teeth packing a fully articulated pop-up espresso machine into a svelte, feminine attache case—on his seventh of fifteen interviews.)
Judge raps gavel.
Damore approaches bench, profusely thanks his lordship, and pretends to forget his fancy coffee on the judge's bench.
Judge: Well, well, look what the bailiff brought me
Judge glowers dramatically at Google's counsel (all twelve), Morse-coding "should I drink this?" with his bushy eyebrows.
One of the sharper members of Google's half-and-half gender-balanced legal dream team (who happens to be a man) recognizes the Morse code gesture (it's a man thing), and Morses back with a precisely calibrated locker-room shrug "go ahead, it's just coffee, we're not quite that petty, you old toad".
That was extremely respectful, all things considered. The legal costs award will barely pay for his morning aftershave, which he seems to consume at twice his previous rate now that Google hires twice as many women, not that it improves his win rate, on either score, not one little bit.