If Damore suspected that women at Google were, on average, shorter than men at Google would you doubt that too, because there is no study on gender differences in height among Google employees?

When thinking about this you should ask what your priors are. If neuroticism is widely, across ages and cultures, found to be higher in women and men, and it is, then wouldn't your prior, without a Google specific personality survey, be that Google women likely have higher neuroticism than men?

To me it seems like an impossibly high standard to expect Damore to do a personality survey of Google to figure out if this oft repeated neuroticism result also holds true for Google employees.

Depends what the implications of that statement were. I think assuming that women at Google have higher levels of neuroticism is fair, but once you start to posit that as an actionable truth is when real information is needed.

Please note the shifting goal posts. At the start of this comment thread you were saying that Damore's memo was certainly unscientific, and now you are arguing that there needs to be a higher bar for action than the fair assumptions that Damore made.

Keep in mind that Damore is a single person giving feedback to a diversity course. He doesn't have the resources, time, or mandate to survey Google's employees to test whether or not this general trend holds. Perhaps, before acting on Damore's recommendations it would be a good idea to do some tests on Google's employees to see if general trends hold true in their specific subset - but that is a really different claim than the initial claim that Damore was unscientific.

If Damore had written that smoking causes lung problems and therefore Google should start taking some actions to stop employees from smoking, it might be merited to do further tests to see if the link between smoking and lung problems holds true for Google employees as it does for people in general. It would not be merited to call Damore unscientific because he had not yet done that research prior to making his suggestion.

It seems like he was cherry-picking studies that fit a personal theory he held before the feedback was ever requested. I would consider that to be unscientific.

If he was cherry-picking studies then for each study he cited there would be multiple conflicting studies. Do you have examples of those?

The fact is that the same study says that men express more psychoticism than women. Do you think he would have been equally railed against if he brought that up in his paper?

Women on average are more socially aware and exhibit higher emotional intelligence than women. There is scientific research to suggest this, and for me anecdotally it also rings true (as a man, but again key word = on average).

Do you find this statement offensive? Should I not say this because it might hurt men's feelings?

Or is it only wrong to say something negative about a race/gender if it implies something negative about certain underrepresented genders and races?

I’m not concerned about being offended or what one should or shouldn’t say. But in general saying I don’t find saying those kinds of sweeping statements to be very constructive, including the positive example.