It depends on your definition of morpheme.
S. Anderson cites an 1880 characterization by Baudoin de Courtenay (Stankiewcicz translation) that a morpheme is "that part of a word which is endowed with psychological autonomy and is for the very same reason not further divisible. It consequently subsumes such concepts as the
root (radix), all possible affixes, (suffixes, prefixes), endings
which are exponents of syntactic relationships, and the like". This is different from Bloomfield's definition as "'a linguistic form which bears no partial phonetic-semantic resemblance to any other form' i.e. a form that contains no sub-part that is both phonetically and semantically identical with a part of some other form". However, in both cases, a root is a morpheme (and a proper name is a root).
The question then is whether there is some way to empirically determine whether a root is in fact a morpheme. This is analogous to trying to determine whether a particular sound is "marked" – how do you determine that one definition is correct and the other is wrong? Indeed, it is not obvious that "morpheme" is a necessary technical concept.
There is no requirement in linguistic theory that the set of roots be closed, though they might be in some language. In English, they class "noun" is certainly open.