Melons pose a Borer problem because their seeds are attached to or embedded in the fruit. There are however permissible methods to remove the seeds on Shabbos.
Cantaloupe and Honeydew Melons
In these melon types small seeds are found in the center cavities of the melons. Because the seeds effectively prevent access to the food (similar to an orange peel) there is a firm basis to permit the removal of the seeds by using a spoon to scoop out the seeds or to shake the mass of seeds off the opened melon. However any individual seeds left remaining upon the melon surface after the bulk mass has been removed cannot be considered to be a barrier to access the fruit and should L’chatchila only be removed by cutting away some of the melon together with the seeds. However, if one does not wish to waste any melon for this purpose one can expel the seeds from one’s mouth after eating the fruit with the seed.
If this is impractical, the seeds may be shaken off immediately prior to eating. If the above two possibilities are too difficult, the individual seeds may be removed by hand or with a spoon immediately prior to eating.
Watermelons
Because watermelon seeds are embedded in the fruit itself they cannot be viewed as being an obstruction to access the fruit. Therefore one should expel the seeds from one’s mouth after eating the fruit with the seeds. {Some Poskim permit one to vigorously shake the watermelon seeds off. However, it is prudent to adopt a stringent attitude with regard to this serious halachic question.}
Shulchan Aruch 319:MB84, Igros Moshe OC:4:74, Shmiras Shabbos Kehilchasa 3:17, Sefer 39 Melochos
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I wonder if it would be ok to cut out the parts of each slice of watermelon that has seeds – this way, you’re cutting part of the oichel with the pesulos