Yes babies can eat watermelon as long as they are ready for solid foods and are at least 8 to 10 months of age.
Babies under 8 months of age should not eat watermelon unless it's blended up but once the baby reaches 8 to 10 months of age the baby can be given slices of watermelon that they can eat themselves.
Watermelon is a pretty easy food for babies to eat and it's healthy for them too.
My kids started eating watermelon at around 8 months of age and they loved it and it was cute seeing them eat the watermelon and having the juices drip down on them.
Make sure to either have their shirt off and have them in just a diaper or at least have them wear a bib while eating watermelon because the juices that come out of the watermelon can get messy.
During warm weather we would sit outside and the babies would be in just diapers and then I would just give them a bath when they gone done eating the watermelon.