I recommend
http://www.physics.ccsu.edu/larsen/tolkien.html for further information.
Especially
http://www.physics.ccsu.edu/larsen/jrrt.html is quite interesting and provides some good quotes.
Clearly, Tolkien was interested in astronomy and so also used it to create a feeling for the reader that the action takes place in a real place.
And so why wouldn't people in M-e look to the stars as well. Why it is wise men...well simply because some peasant from South Gondor doesn't have the education needed to understand what really happens in the sky. You had to have learned something in Minas Tirith's library or in Rivendell.
Now, as far as astrology is concerned...I don't really think so. Of course maybe Malbeth the Seer made his prophecies after consulting the stars, maybe he just took a look at the shape of the clouds and the behavious of chickens as I think the Roman augurs did. As far as I am aware we don't know really.
This however again brings us to that topic I so dislike about providence, fate and luck in M-e. Somehow, it seems however that people there simply had this gift, as Aragorn is said to have gotten it from his mother's side.
So I guess that in Middle-earth prophecies were made without looking at other stuff, but simply be "feeling" wha was going to happen. Maybe some had this power to somehow sense Eru's plan or something like that.
My conclusion: Astronomy - yes, important; Astrology - no, possible