k-means clustered data: how to label newly incoming data
This is not answerable without knowing something about the data itself and where it comes from and what it means.  But it sounds like you're trying to do something awfully dubious.

It sounds like you're trying to come up with labels for free, with a purely automated method.  That usually doesn't work.  There's no free lunch.  You'll need to obtain labels from some ground-truth source of data (e.g., human labelling; retrospective analysis); no amount of pure algorithms can replace that.  See Efficiently labelling training data in machine learning.

Using distance computations might give you a plausible approximation, or it might work poorly.  It all depends on the meaning and distribution of the data.  There will be some datasets where it will work well enough, and others where it will work terribly.  There is no single answer, and no special algorithm that eliminates the need to obtain known-correct labels for your data.