Why didn't Eckhart's character look into his character's future?

I expect that he did. and saw a rosy (for him) future in which Jennings dies at 3:03 at the hands of the FBI, and he (Eckhart's character) goes on to be a big success. (Presumably he didn't look far enough ahead to see war and the end of the world, or maybe he thought that he would have time to build a shelter and emerge into the wasteland intact.) What he didn't know was that Jennings would use the machine after he (Eckhart's character) did, see all the nasty stuff that would happen, and be able to change that future to one in which Jennings survives, the machine is destroyed, and war is averted.

And, of course, Jennings sabotaged the machine right after he was finished with it, so Eckhart's character couldn't see his "new" future.