How do you know it's faked? It may very well be, but at this point the Supreme Court of Honduras, the Congress in Honduras and the military in Honduras all seem to be accepting it as genuine. What evidence, other than the purported author's denial, do you have to the contrary?
Well, yes, they would be considering they were the ones who ousted him.

Given that the purported author of the letter says that it was not genuine, that would argue that he either did not sign such a letter, or that it was signed under duress. Either way, it would indicate he did not resign willingly but was forced to, as would appear quite obvious by the expulsion. It's a bit much to believe that he left willingly and then changed his mind.

"Shutdown of opposition media?" What opposition media are you referring to, because I can't find any stories from reasonably reputable sources. In the meantime, see this.
RSF lists these.

Soldiers went to the headquarters of the daily El Tiempo and the TV station Canal 11 in San Pedro Sula on 28 June and insisted that the journalists stop relaying information coming from members of ousted President José Manuel Zelaya’s government,

Several journalists with the international TV station Telesur and the daily La Prensa have talked of censorship by military officers who asked them to “moderate” their coverage. Soldiers also threatened to arrest them if they continued to broadcast footage or print photos of the demonstrations in support of Zelaya.

Around 10 soldiers stormed into the Marriot Hotel in Tegucigalpa on 29 June as foreign journalists were transmitting footage of a demonstration from their room. The soldiers arrested Argentine journalist Nicolas Garcia, Peruvian journalist Esteban Felix and two Nicaraguans who work for the Associated Press as assistants. They were taken to the Immigration Bureau where their visas were checked and where they were released an hour and a half later after explaining they were journalists. Adriana Sivori, Maria Jose Diaz and Larry Sanchez of Telesur were also detained and then freed.

Two TV stations, Canal 66 Maya TV and Canal 36, were ordered to stop broadcasting on 28 June without being told when they could resume.
Some day, those in this country who wish to do that will learn that it makes absolutely no difference in the way we are perceived by the rest of the world. Nor will it change the fact that we are both envied and despised by much of the world.
So, just out of interest, should we close down the State Department and remove the United Nations from U.S. soil? Are the billions spent in those areas a waste of money?