Watermelon plants need lots of space to spread. For an early variety, like 85-95 day, a plant spacing of three feet apart in rows 8 feet apart should be sufficient. For hill planting, place the plants in a grid of squares at least 6 feet by 6 feet. For the later, larger varieties (over 100 day), plant the starts 5 feet apart in rows 10-12 feet apart. For hill plantings, use a grid of squares at least 8-10 feet each way, placing a plant at each intersection.
This is best done in the early spring as soon as all danger of frost has passed. Your plants are rather late, so you'll be on the line. Most watermelons take more than 90 days from transplant (to distinguish from by seed) to harvest, so unless you have an early (small) variety, you may have to protect the plants from frost at the end of the season.
Watermelons do not like being transplanted, which is why the seeds are usually started in peat pots, or direct seeded under protection. When you transplant, get as large a root-ball as possible with each plant, and be careful not to break them. Keep the plants quite damp until active growth is apparent, after which you should keep the plants well watered, but not constantly damp, to prevent disease spreading. If you plant on black plastic, however, the plants will not have soil contact, and this factor won't be an issue.
Watermelons like a very high amount of humus (decomposed organic matter) in their soil, so if your soil is deficient, you will get much better results by creating a comfort zone for each plant before planting. Dig in a good amount of compost, or even mushroom soil is better than nothing. If the top 6-8 inches is half soil, half compost, that is good enough.
Watermelons (being a tropical plant) also like a lot of heat. High heat will cause faster, healthier plant growth and earlier fruiting, which you will want at this point. I usually use 10 foot wide 6 mil black plastic sheeting, secured by earth staples. Do not use black landscaping fabric, or other products that let train through. Weeds will grow through these, and the soil underneath will crust and harden. Also, If you put down the plastic when the soil is moist, it will not evaporate, and the ground will stay soft and moist all season, so you won't have to water often. Also, mud will not splash up onto the undersides of the plants' leaves, and this helps in disease prevention.
The plants that are growing half an inch apart should be treated as one plant until the plants become established in their new location. Then you can take out the weaker plant by cutting it off at ground level. Do not pull it, as this will further disturb the roots of the remaining plant.
They are also fairly heavy feeders, and should be fertilized regularly with a balanced fertilizer high in potassium and nitrogen. (unless your soil is high in nitrates, in which case lots of nitrogen added will be detrimental. Watermelons like a ph of 6.5-7, but if all the other factors are close to ideal, they will put up with slightly lower/higher ph levels (like 6-7.5).
If you have any further questions as you grow these (or any other plants, for that matter), be sure to ask. Prevention is always better than cure.