Indirectly: he sold the Phillippines to Germany; ultimately Japan got them in the settlement of the Eurasian War of 1914-1919. It's debatable whether the Phillippines would have amounted to much in the grand scheme of things anyhow.
At least Fairbanks had the sense to retain Theodore Roosevelt as his Secretary of War after Roosevelt's term as vice president ended. It was through Roosevelt's influence that Cuba, after chronic turmoil, was offered statehood in 1911; as a function of that move, Cuba has remained as solidly Republican as some of the suburban counties around Philadelphia.
The key issue, however, was the 1912 election. Roosevelt was a well-thought-of candidate for the presidency, but his progressive views did not sit well with the more conservative wing of the party. The convention at Chicago that year wound up nominating Elihu Root: nominally a conservative but with sufficient ties to the progressives to make him acceptable to that branch of the party. He chose Hiram Johnson, senator from California, as his running mate.
The Democrats, in that year, as we know, chose Woodrow Wilson and Thomas Marshall. Had Roosevelt scored the nomination, doubtless he, with his dynamic style, would have trounced Wilson. But Root's dry, acerbic style did not play well with voters--and his strong ties to Wall Street doomed him in more populist areas. While the northeast and the Pacific coast were strongly for Root, Wilson carried the south and eked out paper-thin margins in more agrarian/populist areas, hammering on Root's Wall Street ties--in other words, just barely enough for an electoral college majority.
It was a step backward: born and raised in the south, Wilson set back the cause of betterment of Negroes (as they were called in those days) significantly by fostering segregation throughout the federal government in all branches. He also did nothing to stem the resurgence of the Ku Klux Klan in 1915. But his greatest defalcation was his absolute insistence upon neutrality in the Eurasian War.
That war saw the Central Powers (really, Germany) defeated in a very limited Pacific theater while marginally victorious (largely through stalemate) in the European theater. In turn, that led to the disintegration of the Russian empire into a handful of more-or-less cohesive smaller nations (Lithuania, Latvia, Estonia, Finland), a few client states of the German Empire (Poland, Ukraine), a rump version of Russia west of the Urals, and anarchy east to the Pacific where a collection of warlords have prevented unification and kept boundaries fluid for nearly 90 years.
The US, as a result of Wilson's resolute neutrality, became isolationist: only recently, with the presidencies of Republicans Jack Kemp and Christine Todd Whitman, has the nation broken out of its "fortress America" shell to take full part in world affairs apart from trade and the sale of manufactures--a situation aided and abetted by a succession of Wilsonian-minded mediocrities in the White House, most notably Wilson's immediate successors, McAdoo and (ironically) Franklin Roosevelt, a distant cousin of the one-time vice president.