People who were not familiar with the series were seeing it as another case of casting a caucasian actress to play a non-caucasian role. Johansson's casting was during the height of Asian American activism (late 2015, with the elections coming up, Fresh off the Boat taking off, Doctor Strange and Aloha's whitewashing casting; combined with Johansson's popularity, these propelled this casting into every single mainstream outlet.) They even set up a panel with Asian American actors to discuss/protest it.
The issue with this is that it does not take into account the actual setting of the GitS universe, or the fact that it is a foreign property. The GitS universe has a very different political dynamic, with 2 more world wars than in the real world, a split Imperial Americana, and a rising Japan with refugee crisis and Russian population (from the ward). Combined with the Major's own synthetic body and the series's take on transhumanism, this makes it a very different case than that of Aloha (where a caucasian actress in Emma Stone is cast as an Asian American character) or Star Trek Into Darkness and Doctor Strange, where Asian (non American, but already integrated in American media) characters are simply cast with caucasian (and non American for the former) actors (Benedict Cumberbatch as Khan and Tilda Swinton as the Ancient One). The fact that all these backdrops were not in the 1995 GitS film, which is the most popular one, those unaware of other material will still make claims without knowing the backdrops, but with the "I know the property" badge.
Amongst the likes of those leading the protest were Constance Wu, Chloe Bennet, Ming na Wen, and George Takei, arguing that it is unfair for Asian American actors to not let them have the role. But because this is on the basis of casting, it does not take into account of the cultural translation or in-universe politics (neither of which is a barrier for American properties mentioned beforehand). Chinese actress Joan Chen argued for the former, bringing up the likes of The Magnificent Seven (the American adaptation of The Seven Samurai; Samurai roles are changed to Cowboys and the Meiji setting is moved to the American Wild West, but the general plotting remains the same) and that a case like GitS resembles a translation for American audience, rather than another whitewashing case where an Asian character already in a culturally American setting is replaced by a white character or straight up yellowfaced. Mamoru Oshii himself (director of the 1995 GitS film) argued for the latter, saying that there is no basis that the Major's body must be Japanese. Do note that neither of the latter figures are American, and that Oshii has made a live action movie Avalon, and casted a caucasian Polish actress for the lead role with the intent of her looking like the Major.
That is not accounting for the language barrier (which the film tastefully addresses imo). Casting a Japanese actress in a Japanese setting while retaining an English script is strange and is another rabbit hole to orientalism or rejection from Japanese audiences. Netflix's Marco Polo does this and was ill recieved by both Chinese and Mongolean audiences (not accounting for the cross casting; many Chinese actors were playing Mongolians, while the few Mongolian actors on the casted were quickly killed off). Amazon's the Man in the High Castle does this as well, but is very inconsistent in that Japanese dialogue is in the series, but not consistently with Japanese characters. The majority of scenes with Japanese characters speaking alone were still in English. For GitS, which chose to keep the setting in Japan, casting a caucasian actress to allow the majority of the script to be in English while not eliminating Japanese dialogue is a plus in my books.
Similar cases then popped up with Netflix's Death Note and the recent Hellboy reboot. Ed Skrein was cast for a Japanese character, but dropped out after backlash. A win for Asian American actors, but the actor who was cast after was Korean. Had this happen with GitS it would've been way worse, in spite of being championed by Asian Americans, because the IP was Japanese, and cross casting is usually frowned upon there especially when done by an American studio (see: Memoirs of a Geisha, Tokyo Drift), so this rules out the option of casting a famous Chinese actress then banking it on being a big hit in China, not to mention China is depicted on the losing end of the superpower spectrum in-universe. Death Note had a similar hurdle, except it moved the setting to America, hence it would fall under a case of adaptation and is free to cast any American actor, caucasian or not, except then an Asian American actor was rejected from auditioning for the already American role, which muddles the water even more. Death Note is the case where everyone sucks imo, the claim that an Asian actor must be cast because the IP was Japanese does not work because they chose to move the setting to America (and they should be free to do so. It is their adaptation), while the rejection of an Asian American actor for an American role is also a dick move and is part of what has perpetuated whitewashing for years.
Ultimately, this issue is much, much more complicated than either sides on the political spectrum will tell you it is. Ghost in the Shell had its own set of things to overcome and unfortunately it was caught in the crossfire of everyone else's hurdle on top of it. Its case never fitted the strict, unpoliticized definition of whitewashing (neither was Death Note), unlike Aloha, nor were there any intent from the creators to block Asian actors, unlike Death Note (they sought out Takeshi Kitano, and gave him 2nd billing, and approached the rest of the cast internationally and acknowledge it in the story. A French actress for the Major's creator was specifically chosen to mirror one of the oldest sound recording in human history, the French Au Claire de Lune lullaby (also featured in the film), which can be seen as the first piece of digital information ever. Johansson is the only American member of the cast. But ignoring all that, it can indeed be seen as a would-have-been an opportuinity for an Asian actress, and so that's where the majority of the sentiment came from. During the height of the backlash, you will see many on both sides ignoring and being insensitive to the other. Many Asian Americans were claiming that the Japanese opinions (and Oshii's) do not matter because this is a American adaptation and only concerns Americans (despite the film getting an international release and being based on a Japanese IP). Then many defending the choice were simply downplaying the underlying problem with Hollywood's casting and simply heralding an ideal "Johansson was chosen simply because she was the best actress in the audition" (there was no audition. The director has stated no one else was considered for the role after Margot Robbie turned it down for Suicide Squad). Then there were many Japanese netizens expressing their opinions that it's better that the character is turned white rather than to have someone not Japanese pretending to be one, despite the role being completely synthetic (although admittedly, it is socially Japanese). All arguments equally ignore the context of the franchise and barely anyone has ever backed down from their own stances. Constance Wu, when faced with Joan Chen's argument of it being an adaptation, basically gave a "well yes, but I'm still right" as an answer.
Then there are people who were familiar with the materials and agreed that it is plausble/not improper to cast a caucasian actress for the role, but simply do not like Johansson as a choice on the basis of acting, in which case there is not much to be said as that's a matter of opinions.
That's just my take on it all. Sry for the long post. Welcome to the GitS franchise.