The official words are "I pledge allegiance to the Flag of the United States of America. And to the Republic for which it stands, one nation, under God, indivisible, with Liberty and Justice for all." The existence of the Pledge of Allegiance on Omega IV is likely an example of Hodgkin's Law of Parallel Planetary Development or as the result of a timeline corruption resulting from the Temporal Cold War.
"The Omega Glory" makes it clear that the war there occurred generations ago, and post war the life span lengthened to about a thousand years. This places the war well before 1200 AD, meaning that Omega IV had the United States of America, the Pledge of Allegiance, and the United States Constitution several centuries before Earth.