I didn't hear the Hannity show, and I know nothing about this Navy chaplain.
But I will tell you all this: his complaint is very, very real.
In the Air Force, staff meetings used to be started with a prayer if the Commander so deemed. Catholic, Jewish, and Protestant chaplains rotated the duty.
In the aftermath of the Academy complaint by a radical chaplain Captain (who was 54 years old--her name is Capt. Melinda Morton and she went straight to the NY Times with her complaints) the military has gone overboard and has banned Evangelical and Protestant chaplains from praying in Jesus' name at any public event.
Capt Melinda Morton
Chaplains may still pray in Jesus' name in the base chapel during services, but if they do it outside of chapel (a graduation, a hospital, etc), they are rebuked.