Here is the passage again:
Ralph spent all of his savings on a diamond ring to give to his girlfriend
Melanie. A greedy young woman who had a box full
of expensive trinkets from ex-boyfriends. Melanie was always looking
for the man with the fattest wallet.
The second part is an appositive fragment. All you have here is a noun phrase renaming Melanie. This phrase does not have a main clause, the necessary component of all complete sentences.