Jan:
Here I am reading your question about seedless watermelon, and it's November -- long after your question was asked. But there might be others like me who just located this site and are interested in melons.
A little bit of genetics is helpful in explaining this one. Watermelon, like most living things, get one set of chromosomes from each parent. The chromosomes contain all the genes needed for it to grow, and a gene is just a part of a chromosome that tells the plant how to grow. In people, for example, you might have a gene that told your body to grow blue eyes and another gene that controlled you height.
Back to watermelon. Regular watermelon have pairs of chromosomes (a plant with pairs of chromosomes is called a diploid in genetic terms). To get a seedless watermelon takes several steps. A young regular watermelon plant is treated with a chemical called colchesine (I might have spelled that wrong, but it's close). This treatment doubles the chromosome number so now that melon plant has its chromosomes in groups of four instead of two. Another normal watermelon (diploid) with pairs of chromosomes is now crossed with the plant with the chromosomes in groups of four. Remember that the watermelon will get half the chromosomes from one parent and half from the other. So the seeds in the fruit from this cross will get 1 set of chromosomes from the normal plant and 2 sets of chromosomes from the treated plant. The result is seeds with 3 sets of chromosomes. This seed is the seed used to grow seedless watermelon. With 3 sets of chromosomes (called a triploid), the watermelon is functionally sterile -- the pollen doesn't work right, it is sterile. Pollen from another normal (diploid) watermelon plant is needed to pollinate the triploid female flower. So now you have a normal (diploid) plant crossing with one that is triploid. Because the three chromosome plant and the two chromosome plant are trying to combine, the seeds do not form (except for the odd one) and you get a seedless watermelon. This technique was developed in the 1950s by a scientist in Japan.
The bottom line is -- you must have two types of watermelon to grow seedless ones -- a triploid and a diploid. If you grew just seedless watermelon seed, you would not be able to produce a watermelon. Along with the seedless watermelon plants you must grow a normal watermelon plant to pollinate it.
Sorry for the long explanation. It is not a simple process, but that's how it works!