Welcome to the forum. It is better if you update your profile with your location and zone. It makes it easier to answer questions.
Those are leaf miners an usually are not a big problem unless there is a lot of damage to the leaf surface. It is from the larvae of a small fly. They are difficult to control organically. All you can probably do is use some kind of barrier netting but you have to make sure none of the flies get inside or hatch.
The plant is small so with so much tunneling it is probably not getting enough photosynthesis going on and it looks like it is in a pot?
Depending on where you live it is kind of late to be planting watermelon and only the ice box watermelon will do ok in a large pot in full sun. Usually when the larvae of the leaf miners can be pinched and squished at the end of the trail. A healthy growing plant will usually tolerate the damage so I usually don't do much about it except remove the leaves that are infested. Organic sprays will not help since the flies don't stay on the plant and I don't like to use harsher chemicals on plants that I eat.
Happy gardening in Hawaii. Gardens are where people grow.