The question may sound philosophical and may be subject to heated debates, but to understand this question one needs to look at the structure of Korean society in a sociological, anthropological and cultural context.
So what is a Korean. Or more precisely who is Korean?
Legally: a Korean is the holder of a Korean passport regardless of place of birth, descent, heritage, culture, ethnicity etc. Some people are accepted by society as being Korean but are not Korean citizens (naturalized Korean citizens who immigrated to foreign countries, Korean born abroad to Korean parents but who moved to Korea at a very young age, sometimes as young as a few days old - some parents give birth to their children abroad so they can get foreign citizenship). Some are Korean citizens, but Korean society does not accept them as Koreans.
Sociologically: a Korean person is someone who does not deviate from the social non-written norms that Koreans acquired since childhood. Education, written documents or any form of brainwashing can not change this perception that Koreans have of what a Korean is.
A Korean is in sum, someone who has physical features that are accepted as Korean (people with physical features deviating from norms will not be considered Korean no matter how culturally adapted he may be). It is also someone who has perfect knowledge of Korean language, that is, who speaks Korean with an accent, words and grammar that are accepted as being Korean. So far, anyone who looks physically Korean and who speaks Korean is Korean. But there's more to that.
A Korean is someone who has the linguistic competence of a Korean. That is, a Korean should know what is acceptable to say, what is not, when to say what, what to tell a person he meets for the first time, a taxi driver, the President, using accepted norms and rules for such use of the language. Any deviant use of this kind of language and the person is not considered Korean.
In addition to physical features, language and linguistic competence, a Korean is someone who makes accepted use of body language. Koreans have distinctive ways of walking, reacting to shocking news, and use different body language to show anger, fatigue, despair or pride.
Being Korean is also making accepted uses of the social etiquette. Etiquette can not be put on paper, as there are various variations of single norms which are accepted. But any norm that does not correspond to those variations exclude the person from being Korean. Etiquette includes dress code, handling objects, and subtle things like eating food or drinking coffee, and even lighting a cigarette or taking notes in class.
Note that there is not one single norm for language use, physical features, linguistic competence or etiquette. The exact same rules can have various variations (in opposition to law where often one aspect has one single law), but those variations are finite, and anyone who does not belong to one of those variations is considered deviant, and society questions his being "Korean", or belonging to the nation.
Note that all people belonging to that social definition of Korean are Koreans, or "mirrors" of society. Those who have the physical features of Koreans but deviate in some way, whether linguistically or in terms of etiquette in a blatantly obvious manner (someone who looks Korean but as soon as you talk to him you realize that he's not) are called "hidden immigrants" of society. Those who observe all the linguistic and cultural norms of society but have foreign physical features are called "adopted" Koreans. Finally, those who neither look Korean nor act Korean are called "immigrants". Except for "mirrors", others are not accepted as fully being part of Korean society.
The government may take as much action as it wants regarding the promotion of minorities, it will not change societies perception and exclusion of those who society does not accept as Koreans. That is, the government may criminalize some types of behavior towards minorities, promote the image of minorities, but it can not tell society to treat foreigners the same way they treat Koreans. Such behavior is encoded in the brain, ask a neurologist or a psychologist, and they will tell you that neurons, hormones and what not interact and react in uncontrollable ways when human beings see someone "different".