There is a "classical" approach to hypothesis testing that uses only critical values and never looks at the p-value. There is general agreement among statisticians that the "modern," p-value, approach can provide additional information beyond that of the classical approach.
So, in short, no you don't need p-values. However, modern statistics uses them. In fact, if you go to the New England Journal of Medicine online and look at some of the research articles, you will see something like this: (P=0.004). I can't recall having ever seen (T=2.623 < t_0=3.015).