Watermelons are thought to be native of the Kalahari desert in Africa, due to David Livingstone describing them as "abundant" in that region. One can find, still today, plants of what is believed to be the ancestor of watermelons, called Tsamma melon (Citrullus lanatus var tastius). The Tsamma melon can also be found in Baja California (it was brought there by human intervention, though) and African regions such as Zimbabwe, where it is known by the name of "mwiwa" (or "iswe, nwiwa"): Tsammas' flesh is similar to watermelons, and they have pinnatifid leaves. They are used in Africa to make jams due to their high content of pectin.
Watermelons have been cultivated at least as early as 2000 b.C (there's evidence in the Nile Valley, as noted by Zohary and Hopf). Several watermelon seeds were found in the tomb of Pharaoh Tutankhamen, as well as in other sites of the 12th Egyptian Dynasty, even though no literature nor hyeroglyphics sources depict or describe watermelons.
It appears this fruit was largely unknown in other Mediterranean cultures of the time, but they were certainly cultivated in Asia (China in particular) by the end of the 9th century A.D. The fruit was introduced to Europe by the Moor invasion in the 13th century, and the word "watermelon" was present in English dictionaries in 1615, according to the Dictionary of American Food and Drink by J. Mariani.
Several other Asian regions have cultivated watermelons in ancient history: there are legends about watermelons in Vietnam, for example. According to folklore, it was discovered by Prince Mai An Tiêm, a prince of the Hung dynasty (son of the 11th King).
He was exiled on an island, with the condition that if he could survive for six months, he could go back to his castle: he prayed for help, and a bird flying by dropped a watermelon seed on the groud. When the prince cultivated the seed, he called the fruit dua tây (western melon), since the bird had come from the West. The prince eventually survived and went back home.
According to online sources, watermelons have been introduced to North America in 1500, and Indians were found cultivating them by French explorers in the Mississippi valley.
Other sources believe the fruit was introduced in Massachussetts in 1629, with significant help from African slaves (at least according to John Egerton, a famous food historian).
Jerry Parsons, a Texas Agricultural Extension horticulturalist, credits european colonists and african slaves for spreading watermelons around the world; another historian, Parsons, states that the crop was grown by Native Americans in Florida (by 1664) and the Colorado River area (by 1799). Other literary sources cite watermelons in the Midwestern states (1673), Connecticut (1747), and the Illiana region (1822).
Thanks to the advancements in technology, watermelons have been crossbred and modified to yield more fruit and be more resistant. A good example is given by Charles Fredric Andrus, a horticulturist at the USDA Vegetable Breeding Laboratory in Charleston, who produced a cultivar resistant to wilt and disease. His variety has a long shape, with a hard rind that is easier to ship and store, and it was so versatile it could be grown in very different climates, while producing goodo yields and being resistant to the most dangerous watermelon pests (anthracnose and fusarium wilt).