While there have been some accomplished climbers who have done this in the Himalayan peaks, alpine style remains relatively rare in the Himalayas. Most climbing is done in the expedition style with large groups.
Is there a fundamental reason this is so? Can a moderately experienced climber, who might summit solo or in 2-person teams in European/American peaks, attempt the same in the Himalayas?
TLDR: Because its much harder that way and the extra altitude of the Himalayas makes it that much more difficult.
Alpine style refers to mountaineering in a self-sufficient manner, thereby carrying all of one's food, shelter, equipment, etc. as one climbs, as opposed to expedition style (or siege style) mountaineering which involves setting up a fixed line of stocked camps on the mountain which can be accessed at one's leisure. Additionally, alpine style means the refusal of fixed ropes, high-altitude porters and the use of supplemental oxygen.
As the Himalayas are at extremely high altitude where supplemental oxygen is all but required and extra time is needed to acclimatize having a series of camps makes sense.
It has been done but its not the most common option for a reason.
alpine style remains relatively rare in the Himalayas
Actually, as far as I can tell from the current climbing literature, alpine style climbs in the Himalaya are relatively common these days. Expedition style climbs (at least the only ones that attract any attention) are now mostly for paying customers with guides. I suspect there are also still private alpinists doing hybrid climbs in the Himalayas, but they don't get covered much in the climbing press.
The highest Himalayan peaks (8000 m) are roughly twice the height of the highest peaks in the Alps or North America outside of Alaska and the Yukon (4000 m). The additional altitude makes a huge difference: Climbers can typically acclimatize to 4000 m in a matter of days. Acclimatizing to 6000 m takes weeks, and even alpine style climbers generally have to have support teams or porters to get all the food and gear to a base camp where they do their acclimatization. Climbers don't acclimatize to altitudes over 8000 m, they slowly die. Climbers can get fatal altitude related conditions like cerebral edema at 4000 m, but the higher you go the more common these become, even with supplementary oxygen.
Can a moderately experienced climber, who might summit solo or in 2-person teams in European/American peaks, attempt the same in the Himalayas?
This is too broad to be answered because "moderately experienced" is vague. Is a climber who can climb the Hornli ridge on the Matterhorn, Monte Rosa, or the Gouter Hut Route on Mt. Blanc, without a guide, a moderately experience climber? They probably could qualify for a guided climb of Mt. Everest, but they'd be doing it expedition style. How about a climber who's done all the big North faces of the Alps? They might very well be invited on an alpine style climb in the Himalaya.
