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I guess I'm late to the party, but here are a couple of points:

1. Yes, an arbitrary complete pointed metric space $X$ with finite diameter is characterized up to isometry in terms of $Lip_0(X)$. $X$ is naturally isometric to the set of weak* continuous homomorphisms from $Lip_0(X)$ into the scalars. See Theorem 4.3.2 of my book.

2. In response to another comment, the restriction to diameter at most 2, for spaces without a distinguished base point, is natural. For these spaces the Gelfand transform takes $X$ isometrically into the unit sphere of $Lip(X)^*$. A metric space can isometrically embed in the unit sphere of a Banach space if and only if its diameter is at most 2.

1

I guess I'm late to the party, but here are a couple of points:

1. Yes, an arbitrary complete pointed metric space $X$ is characterized up to isometry in terms of $Lip_0(X)$. $X$ is naturally isometric to the set of weak* continuous homomorphisms from $Lip_0(X)$ into the scalars. See Theorem 4.3.2 of my book.

2. In response to another comment, the restriction to diameter at most 2, for spaces without a distinguished base point, is natural. For these spaces the Gelfand transform takes $X$ isometrically into the unit sphere of $Lip(X)^*$. A metric space can isometrically embed in the unit sphere of a Banach space if and only if its diameter is at most 2.