While the majority of top rated answers here describe the grammatical reasons this is the case correctly, they don’t specifically address why it is unacceptable socially (I.e. why it sounds inherently racist to native English speakers) and those that do address this fail to do so adequately.
To understand why this is, you have to understand that it is common (for complicated historical, cultural, and linguistic reasons) for native English speakers to lightly modify grammatical structure in their own language as a form of contempt or derision, especially when said phrasing refers to a group that can be considered ‘other’.
When said ‘other’ is being referred to in racial or ethnic terms, or as in this case, by a nationality that can also double as an ethnic group from the perspective of native English speakers, this practice can result in seemingly innocuous phrases or those not intended to be racist coming off as racist. This is because native English speakers are conditioned to perceive obvious grammatical mistakes from their perspective, especially when race or ethnicity is involved, to be intentional.
Context is also important. Take, for instance, the following example; it would be very uncomfortable in a polite or neutral/serious context such as describing a person to a stranger for a native English speaker to have another native English speaker describe a third person of Jewish descent as ‘a Jew’. In this case, a religion that can also be an ethnicity in proper singular noun form is used as a descriptive adjective.
Referring to said person in such a singular, direct, and obviously strange sounding (to native English speakers, as explained well in other answers) way implies that the speaker is referring to the person’s ethnicity rather than their religious beliefs, that this status of ‘Jew’ is of particular importance, and that the speaker considers that type of person to have negative characteristics because they have not made the effort to refer to the fact that they are of Jewish descent correctly.
So, within the context of a conversation between two native English speakers who are mostly unfamiliar with each other personally, this misuse of grammar strongly implies that the speaker holds contempt for Jewish people and have either subconsciously revealed such or are revealing such intentionally.
An analogous example related to lifestyle would be referring to someone who is gay as ‘a gay’, although because of the increased grammatical incorrectness of referring to someone collectively as a member of a particular sexuality this could be considered a politically incorrect or offcolor joke, and the Jewish example could fly under the radar if the people in conversation are at least acquaintances.
However, both would undoubtedly be considered to demonstrate racism/homophobia, respectively, within the context of a conversation between strangers.