It is highly unlikely that the U.S. Capital could be moved now (or, even in the past as early as the 1870s!) due to the strong establishment of government offices and the growth of the bureaucracy. Think about all the millions of people who live in DC and work in or near DC because of the government and military. And of course it grows every year.
UNLESS, of course, the Capital were to be seriously destroyed. At this point, it would probably take a direct nuclear hit, but we can surely imagine an alternate universe where a Spanish-American War, World War I (or II), or another war with no parallel laid non-nuclear siege to the city and destroyed it.
I have always thought either Kansas City of St. Louis would make the best centralized capital, especially St. Louis (already a major hub for air/rail/road travel, and they already have one well-known monument!) I like to think it would be renamed St. George (rather than Washington...) in the vein of St. Petersburg.