That's an interesting question and one that is pretty straight forward. When George Washington was in his early 20s, he traveled to Barbados with his brother Lawrence to try and help him heal from tuberculosis. The trip sort of backfired, [with George catching small pox, but he recovered shortly thereafter](http://rethinkingtherevolution.com/blog/britishempire]. When he returned to the Colonies the following year, he would never again leave North American soil. The reason why he never left is relatively simple. After he returned home, his brother Lawrence died from his illness, and George inherited nearly everything. He even inherited a position as a major in the Virginia Colonial Militia. Washington would then be involved in the French and Indian War for a number of years before returning home to become a "gentleman farmer." From 1759 until the start of the Revolution, Washington remained in the local area as he tried to make his father's plantation work. Tobacco farming was quite intense work and required plenty of financial management to deal with the fluctuating markets, so he never really stayed away for long.
Of course by the mid 1770s he was involved heavily in the war by leading the Continental Army, and after the war ended he returned to being a farmer. This period is when Franklin, Jefferson, Adams, and other American gentry were off in Europe and making friendships and political alliances. Washington was either too absorbed with leading a war or, when he returned home too obsessed with trying to fix his plantation. His nephew Lund had poorly managed Mt. Vernon while Washington was away, and as a result, it required Washington's immediate attention. He then spent the better part of the decade expanding the house, struggling to get out of debt, and entertaining guests – and he had hundreds of guests who visited him every year.
Of course we all know that he was elected president shortly thereafter, and served honorably until 1797, when he returned home and two years later he passed away. In short, he was too busy with other matters to afford himself an opportunity to go abroad.