Is there something that violates the Navy from posting pictures of fired personal on the front page of NavyTimes?

The Navy Times is not an official US Government publication. It is a civilian newspaper which is is not controlled by the Navy, the Department of Defense, or the US Government. Like any other newspaper in the United States, the paper's writers, editors, and publishers have a Constitutional right to freedom of the press and they can publish whatever they want irrespective of any impacts it may have on the morale of the military. The reporting must be factually based and cannot be malicious. Their desire is to sell more papers, increase the paper's circulation, and get busiensses to advertise to reach the paper's readers. That's how newspapers are supposed to make money.

If you do not like what you read in any newspaper, you can write a letter to the editor and voice your opinions. Most newspapers will gladly publish your letter for free and unedited even if your intent is to be critical of the paper. This can be a good way to let the readers know what you think and to also inform the paper's management and advertisers that you don't like what the paper is doing. Sometimes letters to the editor can cause staff changes at newspapers depending on what the issue is.