This post duplicates my Quora answer to a question: Does IBM Watson have elements of deep learning in its code?. Feel free to vote and comment there.
Let's assume that we talk about the original IBM Watson that won the Jeopardy! and not a series of diverse products collectively branded as Watson. Keeping this assumption in mind: no, I don't think that IBM Watson used deep learning.
At the very least it didn't seem to play any substantial role. IBM Watson seem to be relying on a classic approach to answer factoid questions, where answer-bearing passages and documents are found via a full-text search. A small fraction of answers came from knowledge bases, something around 1%. Another interesting technique (again it didn't account for many retrieved answers) is previously in my blog entry: On an inconsistency in IBM Watson papers.
To select answer candidates, IBM Watson used a bunch of techniques including filtering by an answer type (e.g., for a question "Who is a a president of the United States" an answer should be a person). It also helped that most answers (90+%) were either Wikipedia titles or a part of the Wikipedia title. Additional help came from redundancy in answer-bearing sentences, i.e., an answer was contained in many text passages.
Disclaimer: I have never worked for IBM and my understanding comes from reading several articles that the IBM Watson team published several years ago.