OK. YOU wanted it Style comments are just my personal (not very humble) opinion, so don't take them very seriously.

they looked totally alien to his native California... particularly the ones with white bark stained with horizontal strokes
I don't think this sounds quite allright. Perhaps, 'they looked totally alien to the trees he was accustomed with back in California'... not the best variant, but here trees are supposed to look like an US state
Anyway, native California is simply wrong (or so I think). I know of course what you were trying to say, but I'm sure the word 'native' cannot be used this way. Try 'home in California' or smth.
Horizontal strokes?

The weather was surprisingly fresh, even cold. It felt like it was early summer
surprisingly? Or it felt otherwise?

It didn’t feel like amnesia
Memory loss?
Here and further. In Russian fiction I noticet much is said from a neutral point view (it felt like, it appeared, etc), but in English fiction things are more personalized. You find more 'He/she felt like, he/she saw smth appeared, etc).
So, I suggest you write this 'He didn't feel like he'd lost some part of his memory'.

Peter had been spending time entertaining himself with parties, safari tours and women
Suggestion: spending time for amusements like ...

always craving for attention measured in dollars and euro
Quite unlikely that the Californian girls even know what Euro is. Perhaps just 'money' will do.

She truly believed that her best friends were diamonds, not people.
Буээээ. Bad, bad, bad cliché.

a huge house with all those green terraces
But all of it altogether didn’t make him feel happy, topped up by his constantly deteriorating health, which would go worse and worse from year to year.
But nothing of it made him feel happy not to mention the slowly deteriorating health that didn't promise to be any better in the years to come.

Looking back at his life he realized that there arewere a few things money can’t could not buy. Without them life becomes pointless
Without what? Things money could not buy or the money itself?

He tried almost everything – gambling, hunting, fishing, sailing, diving…but nothing brought him the relief he was longing for so much.
Almost everything you say? (if you know what I mean, surely you can invent some more 'exotic' distractions for an old rich fart.
Relief? Satisfaction perhaps?

That's all for now. I'll continue later.