I just found the question, so the answer might come a bit too lat, but.. Have a look at:
Paweł Waszkiewicz, "Common patterns for metric and ordered fixed point theorems.", In Proceedings of the 7th Workshop on Fixed Points in Computer Science (Luigi Santocanale ed.), 2010, pp. 83-87.
I attended this talk last summer, and it addresses exactly your question.
As suggested by Jacques, I turn my comment into an answer.
This is not exactly what you ask for, but it is related. Efe Ok in Section 3.4 in Chapter 6 of his yet-to-be-written book on ordered sets gives a proof of the Banach fixed point theorem using the Kantorovitch-Tarski fixed point theorem: