Adrian Veidt is the smartest man in the world. He realized three years before killing half of New York city that the world would never reach world peace at the direction it was going with all the advances in technology and politics. He needed to strategically take all of his former teammates away from the equation. He did this because they were the only potential threat to his plan of eventual world peace.
First Veidt Taking out the Comedian; the first person to realize his goal and also the person most likely to stop him from destroying his way of life. The Comedian had this sense of truth to him because he didnt care for the conventional all American image. He understood that the world was a cold and cruel place. Those that understood that were the ones that would survive. The very thought of world peace destroyed his entire way of life. It was only natural that Viedt murder him first because the Comedian had the most to loose.
Second was the perfect man Jon, whom was made of pure existence. Jon was also the United States only blockade against outside countries willing to over throw them with nuclear weapons. So basically Jon represented the only block between a war, take him out and tension will arise.
Next he framed the only active member of his former team still at large, Rorschach for murder and thus taking out the person most likely to put pieces together and discover something wrong, and of coarse Rorschach did put them together. Before being framed for murder and disobeying the Kleen act (an act that outlawed vitualantism among normal citizens).
What Veidt did not take in to account was two former teammates involvement, Nite-owl and Ms Jupiter. Rorschach warned them all of suspicion and as time when on the two remaining members untouched by Veidts plan, broke Rorschach out of jail and discovered the plan of Veidts to engineer a war.
As all four Heroes rush to Veidts base of operations they learn that he has already killed Half the city claiming the cause to be of an alien attack. This in turn stopped the war from happening and unified the nations of the world against a threat much greater than any of them. The worlds last true hope for world peace yet if any of Veidts former teammates were to expose him the reaction would be immediate war and world destruction.
The only team member not willing to bite his tongue is Rorschach, who before attempting to stop Veldt, sent a complete journal of his plan the citys newspaper. Rorschach was killed because of his potential to leak Veidts innocence and destroy the world but the journal was mistakenly printed days later.
In the end the fate was the same despite the manipulation of Veldt and the silent voices of his former teammates. If anything Veldt accelerated the destruction of the world.